So as usual, it's up to your friendly neighborhood Noah to settle the debate once and for all. *chuckle*
Here's the thing. Calling anything "capital" at all requires a simplification and abstraction. A drill press is different than a building, an oil field, or a computer. Lumping a bunch of stuff in together, putting a dollar value on it, and calling it "capital" is a huge abstraction. This was pointed out in a famous debate called the "Cambridge Capital Controversy." Well, folks, that's how modeling works. Any time you make a model, you make simplifications and abstractions.
Human capital, no matter what you call it, is different than other kinds of capital. It's different in the way it's produced. It's different in the ownership laws applied to it. It's different in the way you extract value from it (in the costs of extraction, how it enters into production functions, etc.). It's different in the way it depreciates with time and with usage. Etc.
Lumping human capital in with other forms of capital requires you to take a stand and say "I don't think those differences are important, at least for the phenomena I'm trying to model right now." Other times, if you think the differences matter, you'd keep human capital and other capital separate.
Economists do the same thing with consumption. In models where economists think the main important feature of consumption is its timing, you lump all consumption together if it happens during a certain period. That's where you get your "u(c)" in macro models. But if you want to model the consumption of, say, peanut butter and jelly, you might separate your utility into u(c_peanutbutter, c_jelly). Etc.
There's nothing wrong with this, per se. You can make stupid assumptions, of course, but that doesn't mean all simplifying assumptions are stupid.
So how should we think about human capital? Here's an analogy that I think works well. You agree that a chainsaw is capital, right? OK, now imagine a chainsaw that you graft permanently onto someone's arm, like Bruce Campbell in the movie Evil Dead 2. It's so thoroughly grafted on that you can't remove it without making it permanently useless.
This chainsaw is very very much like human capital.
Like human capital, the arm-attached chainsaw requires resources to create, including the resources of the eventual owner (he has to hold his arm still, at least, and spend some time undergoing the grafting procedure). Like human capital, you can use the chainsaw to create future value - for example, you can use it to chop up skeletons, demons, and other baddies, like Bruce Campbell does in Army of Darkness. Like human capital, creating value from the chainsaw requires the owner to sacrifice some leisure. Like human capital, the owner can rent the chainsaw out, but he can't sell it to anyone.
(The main difference between the chainsaw and human capital is depreciation. Skills often increase as you use them, while the chainsaw will eventually wear out from chopping up baddies.)
So if you think a chainsaw is capital until you graft it onto Bruce Campbell's arm, but then suddenly becomes non-capital, fine. But now the ways in which human capital acts like other forms of capital should be clear. (By the way, if you think this example is fanciful, watch this video.)
Here's another analogy that I think is useful for understanding the difference between "capital" and "labor". It's a finance analogy. "Capital" is an option (which gives you the right to extract value from something), and "labor" is the exercise fee for that option. "Human capital" is an option you can't resell - the only way to extract value from it is to pay the exercise fee (the labor). "Physical capital" and "land capital" are options you can resell.
Therefore, whether human capital is really capital depends on what decisions you're trying to model. It might be, or it might not be. If you're trying to model a company's decision to invest in worker training, and the workers have lifetime employment, then you probably can go ahead and model human capital the same as other capital. If you're modeling a country's decision to invest in education as a development strategy, you can also probably treat human capital as capital. But if you're modeling people's decisions to get PhD's, then you probably shouldn't model human capital the same as other capital.
For some applications, actually, you can actually represent anything as capital - just calculate its expected present discounted value, and voila, you're done.
So what about the moral dimension of human capital?
If our social welfare function cares about wealth inequality, should we count human capital as wealth? Well, I think it depends on that exercise fee - on the disutility of labor. Suppose I really love writing silly blog posts, and I know that people will always be willing to pay me to do it. In this case, my blogging skill really is a kind of wealth, because since I love doing it anyway, the exercise fee is low. But suppose I also had coding skills with which I could make money, but really hated to sit around coding. Well, in that case, the cost of extracting value from my human capital would be very high, and it wouldn't really represent much wealth.
Some people oppose the use of the term "human capital" because they think it allows conservative types to claim that wealth inequality isn't as severe as it appears, since poor people have human capital. Actually, this is wrong - if you count human capital, wealth inequality will be much much much worse. Rich people have a lot more lifetime earning potential than poor people, and their work is probably more pleasant too.
Other people oppose the term "human capital" because they value leisure as a special good. If I own physical capital I can resell my capital, and have all the leisure I want. But if I have human capital, I have to give up leisure to get value. The more our social welfare function values leisure relative to other things, the less human capital adds to welfare.
You are, of course, entitled to your own social welfare function, so you can care about anything you darn well please. And you're also entitled to your own modeling conventions and definition of terms. So whether human capital is capital is up to you.
Update: One more objection to the use of the term "human capital" is that it objectifies people - it seems to imply that human beings can be bought and sold (even though this is not actually the case, as the chainsaw analogy demonstrates). In fact, "skills capital" would be a better term - especially because in the future, AIs will be able to learn skills too. One great thing about economics is that you can make up and use your own terms. So I say, if you don't like "human capital", use the term "skills capital" instead. There's really no reason not to. Maybe it will spread.