You've pretty much precisely nailed this.

If you go to the classical economists (Smith, Ricardo, Mill, Marx), you'll find that they tend to discuss both specific classes of goods, and specific factors to the wages of labour.

Of the first, from Smith, generally: commodities, wages, rents, interest, assets (gold and silver), capital (stock), and public goods ("expenses of the sovereign"). These are somewhat scattered about the book, the first six largely in Books 1 & 2, and Book 5 devoted solely to the latter.

I'll turn it over to Smith for the factors in compensation of labour, as he's uncharacteristically direct and brief:

"The five following are the principal circumstances which, so far as I have been able to observe, make up for a small pecuniary gain in some employments, and counterbalance a great one in others: first, the agreeableness or disagreeableness of the employments themselves; secondly, the easiness and cheapness, or the difficulty and expense of learning them; thirdly, the constancy or inconstancy of employment in them; fourthly, the small or great trust which must be reposed in those who exercise them; and, fifthly, the probability or improbability of success in them."

https://en.wikisource.org/wiki/The_Wealth_of_Nations/Book_I/...

(An expansion of each of the five elements follows.)

From Ricardo we get the Law of Rent and Iron Law of Wages (Lasalle, Marx, and Engles also discuss this). The first notes that for any productive input whose quantity is fixed, the price will rise to capture all of the generated value. The second notes that wages fall to or below the level of bare subsistence.

The second factor is most distinct for nondifferentiated labour, that is, unskilled, and in particular, in a stagnant or declining economy. Smith describes the circumstances of the latter in terrifying detail in Book 1 Chapter 8, which I strongly recommend reading in full, though the relevant detail begins at: "But it would be otherwise in a country where the funds destined for the maintenance of labour were sensibly decaying..."

https://en.wikisource.org/wiki/The_Wealth_of_Nations/Book_I/...

Smith, et al, are classical economists, and their observations were given mathematical rigour and explanation by the marginalists. Effectively, goods producing rents have zero price elasticity, public goods have zero marginal cost, and in the case of wages, there are generally distinct short-term and long-term cost components, a phenomenon which can produce a backwards-bending supply curve. That is, if you reduce a wage rate below subsistence, on a time-worked or piece-work basis, you will get more productivity as labourers work more hours to simply meet their minimum survival requirements. This comes, however, at the cost of long-term capacity: raising of families, rest and recuperation, education, and the like.

Much of the conflict of modern economic life, from a worker-renter's point of view, can be explained by the squeeze of falling wages against rising rents, via numerous mechanisms.

Your observation on who has rightful claim to the surplus value generated by labour (and the role of increased productivity in setting an upper bound to the possible wage component) is the matter explored by Thorstein Veblen, though I need to revisit just exactly where.