Yes and no. In my location, no this will not alter the title to indicate salvage, rebuilt, or reconstructed or whatever word you like if you keep it and accept the price the insurer offers. Should be nice right now because scrap metal is low, not worth much. Here, we don't have to pay for a new title because one is not issued. It stays the same and is not branded, just "transferred" momentarily. If you sell the salvage to the insurer, the new title is marked then. If you keep it, the old title stays clear and remains in your name. Lots of cars have salvage titles just because people are rich and lazy and believe "throw away and buy new" is the solution to everything. "Hooray" for them. I'm poor and know how to fix cars so I "keep the salvage", fix, and drive or sell with no messy title label. Both my FWD GPs have been a "total monetary loss" twice but both titles are clear.
A rebuilt title is not an indicator of significant damage necessarily, more often it just reflects the low value of the vehicle and that it has needed some repair at some point. Keep that in mind as you shop for a replacement to use while your favorite one is down being fixed.