Price at the edge
Price is a number attached at the edge of a thing. People who treat the number as the whole measurement throw away the axis it sits on, the direction that axis runs, and the market the number was actually set against.

Axis
Value is one word doing the work of several, and people use it for all of them at once.
Pull it apart and it goes like this. Value is the axis: the line a thing sits on. Taste decides which direction on that line is up.
Quality is the distance between where the thing sits and where the spec says it should sit. Cost is what closing that distance takes: hours, attention, capacity, the things a person actually spends. And price is the cardinal figure attached at the edge, the one number in the whole set that somebody can write on an invoice.
Four of those are ordinal. They say more or less, higher or lower, closer or further, and they never say by how much. One of them is cardinal.
It says exactly how much, and because it is the only one that does, it gets mistaken for the measurement itself.
That is the term to carry out of here. Price is not value. Price is what value looks like at the one point where it touches money, and everything upstream of that point is measured in a different unit.

Price
So why does the number matter so much, if it is only the edge?
Because in any system that runs on records, pricing is what makes a thing exist. That sounds like an exaggeration until you have lived the other side of it. Unpriced value is not undervalued.
It is absent. There is no line for it anywhere, so no decision can be made about it, no split can include it, nobody can even be wrong about it.
I spent years explaining what my work cost. Energy, capacity, the attention it pulled from everything else, the load I carried so an arrangement could look like it was running. All of that was true and all of it was ordinal, and I was saying it to people who keep their books in cardinal.
I was describing something that had no entry in any account, and the explaining produced exactly nothing, for years, because there was nothing on the page for it to attach to.
Get this wrong and the loss is not a smaller fee. It is that the thing you did is not in the system at all, and you find that out only when the system makes a decision without it.

Market
The number at the edge has one more property, and it is the one that caught me most recently. A price is set against a market, and the market is not a fact about the work. It is a fact about who is buying.
I had an advisory line quoted in the currency of the place I live. The client was buying from a market where the same work is sold in a currency worth many times more. I found this out the way these things are usually found out, from the first delivery that was properly scoped, when the shape of the work was finally visible and the number beside it was visibly wrong.
Off by most of an order of magnitude. Not because the work had changed. Because it had been priced against the wrong market the whole time.
The value had not moved. The axis was where it always was, the direction was the same, the distance to the spec was the same. Only the edge was wrong, and the edge is the only part anyone else ever sees.
Which leaves the question I still do not have a clean answer to. If the number is set by who is buying, and who is buying is something I only discover after the work exists, what exactly am I measuring when I write a price down before the first delivery?