Lawyer here. The legal answer on where to locate the fence is it depends on where you are. If money and time aren't an object, then pay for a new survey, personally call the local government and ask them if there are any setbacks, and if you are in an HOA check with them too.

Where I am, you are free to build your fence on the property line if you want, however, some people don't, but that doesn't mean you are ceding that property to your neighbor. It's still your property. Adverse possession in most US locations usually takes decades and is more involved than merely using someone else's land, plus it's not cheap to enforce.

I've seen where neighbors don't get along so neighbors intentionally do not build a fence on the property line. Instead they build it a lawnmowers width back, so that they can mow both sides, repair the fence without needing permission to enter their neighbors land, and to ensure that their neighbor isn't getting a free fence (don't want anyone connecting to their fence).

When I was at the DA's office (intern), I had a case I was prosecuting where neighbor #1 pressed trespassing charges on neighbor #2 for crossing the property line to stain a fence. Neighbor #2 had been denied access, so had tried staining the fence by leaning over, but couldn't reach the last few inches. Neighbor #1 came home from work and saw that 3 inch strip had been stained, so he knew that someone had snuck over and finished it up. Ended up dismissing the charges when neighbor #2 promised to not do it again. To be fair to neighbor #1, he lived on a lot that backed up to a store, so everyone used his yard as a cut through and it caused a lot of yard damage. Neighbor #2 for years allowed his kids to use that cut through path despite neighbor #1 asking him to stop them.