You almost had it, but the for profit (in the marxist The Capital) is exactly what you said here:

"But no one (or,** few people anyway**) want money purely for the sake of having money"

That phrase, that "Want money purely for the sake of having money" is the definitive aspect of capitalism.

What you implied the beavers did is a Commodity-Money-Commodity model (edit: money=work realized in case of beavers) and it is what commerce does and how humans lived before capitalism (no sarcasm but humans lived quite well without the machinery of capitalism).

You make it very clear with the phrase "they want money because it allows them to have other things. Food, housing, good conditions for mating and raising their young."

This is C-M-C model, which defines the proletariat.

Now, capitalist (which makes capitalism exist) are exactly the opposite.

They live based on M-C-M model.

They only purchase a commodity with the intent of turning it into a profit.

In short, they use money for the sole objectivity of having more money (so that they can use more money to have more more money).

This is capitalism:

Turning the monetization the end goal and the winner(???) is the one with the biggest numbers.