ARV is the right term. After repair value. AVR is a typo that gets copied around, and AVM is a different thing, an automated valuation model, which is what Zillow's estimate is.
ARV is a conditional number, and the condition is your scope of work. It means: if this house is finished to this described standard, what will a buyer pay for it. Change the scope and you change the ARV, so the honest way to write it down is always as a pair. Scope of work, then the value that scope produces. A house that gets paint and carpet only might come in at 300k. The same house with counters, appliances and updated baths might come in at 325k. Neither number is wrong. They answer different questions.
On who decides, three people have opinions and only one of them has money at stake with you. An appraiser gives a supported opinion, usually subject to completion of your written scope, and lenders on renovation loans normally order one. A listing agent who sells in that specific subdivision will give you a number for free and often a good one, because they want the listing. You produce your own from comparable sales, which means recent sales of similar houses nearby that are already in the finished condition you're aiming for.
The part that catches people from the trades side is that comps have to be sales, not listings. What a neighbor is asking tells you nothing. And in roughly a dozen states sale prices aren't public record at all, so in those places you'll need an agent with MLS access to see them, which is a real reason to build that relationship before you need it.
