OK, so for me "positive cash" and "profit" are synonyms, with "[not] extracted" meaning "[no] dividends".

Positive cash flow and profit are almost synonyms although there can be subtleties they are not relevant to this discussion.

The parent comment is making a common mistake that non-profits can not make profits, that is false. Non-profits can't distribute their profits to their owners and they lack a profit motive, but they absolutely can and do make a profit.

This site points out common misconceptions about non-profits, and in fact the biggest misconception that it lists at the top is that non-profits can't make a profit:

https://www.councilofnonprofits.org/about-americas-nonprofit...

As the government sees it, you realize "profit" when you, as an owner of the business, take the money it makes for yourself.

That's not the case in the US. Depending on corporate structure, if your business makes more revenue than expenses, even if none of it is paid out and it's all kept in the business, you will either owe corporate taxes on that amount (C-Corp or non-pass through LLC) or the full personal income tax rate (pass through LLC).

Not saying you can't owe tax on it but isn't that unrealized profit?

"Unrealized profit" is a term used for investments or assets afaik, when the paper value has increased but the gains haven't been realized by selling.

For a business, revenue minus expenses in a given accounting period is considered profit. The only question is whether it gets treated as corporate profit or personal income.

This is bogus and doesn't even make sense.

That would mean that any publicly traded company that didn't issue a dividend didn't make a profit which no one believes.

Do you really want to claim that Google has never made any profit?

It's all quite confusing. A non-profit can as you say turn a profit but isn't supposed to distribute it to owners.

There is a difference between positive cash flow and profit as profit has differences in accounting rules. If you invest in some asset (let's say a taxi car) today, all of that cash flow will happen today. But there will be no effect on the profit today, as your wealth is considered to have just changed form, from cash into an asset. For the purposes of profit/loss, the cost instead happens over the years as that asset depreciates. This is so that the depreciation of the asset can be compared to the income it is generating (wear and tear on car vs ride fare - gas).