In the Lean world (as in Lean Manufacturing), the definition of value is the customer's definition. That makes sense, as that's the definition that, at least over the long term, controls how much you can make.

Something someone is willing to pay you money for is fairly simple. (It's not all-encompassing, of course.)

So... profit?

Value is closer to revenue than profit, IMO. Profit is an internal accounting issue.

If you give me two $10 bills in exchange for a single $20 bill, it's clear that the value of what you gave me was $20, but unless you are counterfeiting or laundering money, I presume your profit was $0.

Conversely, if you save the seeds from tomatoes, grow them in your yard, and sell me $20 of tomatoes, same $20 of value, and presumably close to $20 in profit.

Value is definitely closer to revenue, but it's important to note that it can diverge. For example, channel-stuffing [1] creates revenue but does not deliver value. Fraud too can have high revenue but deliver no value.

[1] https://en.wikipedia.org/wiki/Channel_stuffing

Goods and services that don't turn a profit can still have great value. For instance, can you monetize the air you breath? Not really, but I imagine it has great value to you.

Dive boat operators and dive shops monetize air all the time.

Except that's clearly not the good or service I'm referring to is it?

Breath is monetized quite often. How much do you think an assisted breathing machine costs? Or a CPAP mask, or a scuba tank?

Everything is monetized, and profit drives everything, we are a capitalist society. A business will not operate on charity very long...