I understand, and construe, the concepts of productive and unproductive labor in Marx in a layered form:
In any society, regardless of its specific mode of production, labor is productive if it results in a use value. Use values are physical goods that meet any kind of "material need" whether it is a "material need" for human affection, knowledge, ideas, food, etc. (more on this below).
The modifier "physical" is meant broadly to include, say, information, knowledge, etc. (which necessarily exist in some physical support medium, e.g., human neurons, books, hard drives, acoustic waves in the air, electrons flowing through a cable or through the air, etc.). Note that this distinction doesn't mean that you have goods that are pure stuff and pure information in another, but it rather cuts across all goods, as they incorporate some information, etc. and exist in some physical form. Another important point about this is that use values include goods that meet "material needs" and are consumed at once as they're produced (e.g., transportation, haircuts, live teaching, live performance arts, etc.).
In a market society, labor is productive if -- on top of being productive of use value -- it is productive of value, i.e., if it results in a commodity. In this setting, labor spent producing goods for self-consumption (e.g., taking care of one's kids or partner, cooking one's meals, growing one's own food, etc.) -- although productive of use value -- is not productive of value. Note that labor cannot be productive of value if it is not productive of use value. The premise for a commodity to exist is to be a use value. This means also that, to the extent there's excess demand for a "commodity," the redundant stuff is not use value proper. For all effective purposes, that labor has been wasted -- it is not value producing because it is not use-value producing.
In a capitalist society, labor is productive if -- besides being productive of value -- it is also productive of surplus value, i.e., more value than required to merely cover the cost of labor ("variable capital"). Here, labor spent producing commodities by, say, a farmer who owns her/his land and other means of production, and works by her/himself, is productive of value and, therefore, productive of use value, but not productive of surplus value or productive in the capitalist sense. Note that labor cannot be productive of surplus value if it is not productive of value and, therefore, of use value. Surplus value is simply value but beyond a point, so it must be use value in the first place.
In a capitalist setting, teachers or hospital workers who work for a public entity are not considered productive of surplus value, since they are not working directly for a private capitalist. But they are certainly productive of use value. Taxes are somewhat disconnected or mediated (e.g., not proportional to use, nontaxpayers may have entitlements, etc.) from the direct use of public goods and thus the distribution of these goods cannot be viewed as a market transaction between taxpayers and public agencies. However, if a public agency sold the good to the public, then the labor of those public employees would also be productive of value (still, not of surplus value).
Again, there's no basis to attribute to Marx the notion that only labor spent producing tangible physical goods (stuff) is productive. Consider software production or the production of science. The main element of the product is not the physical support on which the software or science always have to exist but the intangible information content (abstract code). In Marx's terms, software and science production is "material" production because it meets "material needs." Another example is (male or female) prostitution. It is productive of use value. By freelancers, it is productive of value as well. Under a privately-owned bordello, it is also productive of surplus value.
A frequent source of confusion is the meaning of the modifier "material." In Capital and Grundrisse at least, it is clear that "material" is not opposed to "ideal," but opposed to "social." (Here Marx follows Hegel's distinction between "Materie" and Form in the Logic.) Marx's usage is akin to, say, the distinction made by the sophists between "nature" and "convention," on which all social critique is based. Under capitalism, the material process (use value production) is "nature," but it exists under the form of a social "convention" (surplus value production). Humans can do without surplus value production (if the stars align properly), but we cannot do without use value production.
And by the way, this doesn't mean that conventions are not objective (independent from the mind and will of individuals). Social "conventions" have different levels of objectivity: fashions may not be very hardened conventions, but other social relations (laws, private ownership, etc.) tend to be much more hardened. The objectivity of use values is even more hardened, but not as much as the objectivity of the laws of physics.
An issue that usually leads to confusion when studying Marx is the concept of "material needs" or, simply, "needs." A test to decide whether a particular need in the usual sense of the term is a need in Marx's sense is to ask ourselves the question: Would this need survive in a hypothetical communist society with no markets and no state? (Even if you believe communism is utopia, play along.) If the answer is no, then that it's not a need but a "social necessity". A "social necessity" is a particular society's necessity to reproduce its relations of production and other social relations, e.g., the social necessity to legislate and enforce private ownership laws, the social necessity to trade in a market, the social necessity to make sure workers don't steal time or stuff from their boss, etc. Clearly, all politics and most legal stuff is unproductive of use value and, therefore, of value and use value.
In a market society (including but not limited to capitalism), Marx calls unproductive labor that spent in assisting the "change of forms" (as in C-M-C or M-C-M', i.e., trade). This, however, should not be confused with the labor spent by a Macy's employee wrapping gifts, providing customers with factual information about goods (as opposed to pushing a brand), or by a restaurant's employee serving food (as opposed to collecting the check). Also, in this kind of society, labor spent in enforcing private ownership laws (as opposed to, say, traffic rules, which under communism would be internalized by people as ethical norms) is regarded as unproductive. Similarly, in a strictly capitalist society, Marx calls unproductive the labor spent in policing labor at the work place (as opposed to technically supervising it).
Of course, these distinctions are conceptual and abstract. Turning the concepts into operational measures may be very costly, in which case approximations will suffice. Some employees, as suggested above, perform an array of duties mixing surplus-value (hence, value and use-value) producing tasks with change-of-forms tasks (billing, collecting, advertising, selling, paying, etc.). Still, IMO, this empirical difficulty doesn't mean the distinctions are useless. Most often, I find Marx's distinctions to be more precise than those used in the current economic literature on, say, "rent seeking" or "predatory activities."
A final yet crucial clarification is that, although Marx often uses (as I did above) the figures of speech "past labor," "dead labor," "materialized labor," "labor spent," "labor embodied," "labor contained," etc. when referring to labor that "produces" value, he made it crystal clear that he actually meant "labor required," i.e., the labor required in the future. The labor that "creates" the value "embodied" in a commodity is not labor actually spent producing such commodity, but the labor that at a given point in time -- given the magnitude of the needs or, as Marx puts it, "the stomach of the market" and the normal productive conditions in a society -- is *required* to reproduce that commodity. He wrote that in the formulation of value as "socially necessary labor time," the modifier "necessary" meant "required" in this sense.
I don't think many Marxists realize that the notion of value in Marx is expectational, as in a Bayesian conditional expectation. Since Marx's usage goes back and forth between "labor spent" and "labor required," he seems to have believed that the labor spent in the production of a commodity (assuming needs are given, under existing productive norms, etc.) was a good indicator of the labor required to reproduce it. Marx discussed value in the abstract and he tended to assume "normal conditions" (supply equals demand, ownership laws are enforced, etc.) in such discussion. So, his implicit notion of uncertainty -- at least under "normal conditions" -- was rather ergodic or stationary. Another way to put it is to say that Marx implicitly assumed "perfect foresight" -- i.e., "rational expectations" in a deterministic world.
I may write later a note on the meaning in Marx of "sociological" categories of "worker," "producer," "proletarian," etc., which have been confused a lot. But let me say here that I don't mean to imply that people (e.g., Marxists or philo-Marxists) should stick to Marx's usage -- he wasn't always completely consistent and his terms evolved over his intellectual lifetime as he needed them to evolve. One step of real political progress is much more important than loyalty to old terms, which many become confusing and outdated as the contemporary references are lost to new readers and other things change. The dynamics of human language and scientific terminology is a hardened objective process too and individuals cannot pretend otherwise. Still, it's not a bad thing to straighten these things out philologically, if not for other reason because semantic and terminological discussions sometimes help illuminate substantive issues in the present.
Julio