The reason the valuation is desired, seeming of late, is that they will charge the sales tax on the sale as a part of the change-over. When they started doing that, they would use the used car industry's "Black Book", which is what was usually used by used car dealers to value trade-ins at. As it was a compilation of the region's used car auction prices rather than a "price guide" of somewhat "supposed to be" values. Which also made the sales receipt only good for verifying the owner bought the car from a particular owner. In some cases, I suspect, the purchase price could be for $1.00, just to show that it was bought and not a gift.
But the Black Book is usually only for more late-model vehicles rather than for older vehicles. Which can be a problem. Which probably led to the "appraised value" orientation? A valuation by an "uninterested third party", for a fee. But even this approach might have its pitfalls as the appraiser will probably use dome published price guide, just determining which price level is applicable.
From my observations, the prices in the Black Book were more "laughable" than not, as to how low they were in comparison to other price guides. Contrast that with the prices of vehicles at the Mecum or B-J auctions, which get all of the press and exposure. Which leaves a BIG gap, usually!
To me, a new as-produced vehicle from the 1960s would probably be closer to a #2 condition than a #1 condition, all things considered. Yet it it "runs and drives", it could be an easy #4 condition vehicle or #3 if it's a really nice "survivor" car. Of course, FEW people in any DMV might know these things, which can vary from office to office, too. Which leads back to the "uninterested third party" appraiser.
The OTHER place where "value" is important is in the area of vehicle insurance. Which can be a whole 'nuther barrel of centipedes. When I had to restructure my multi-car policy with a major company, they wanted ME to put an estimated value on each of the cars . . . as they all were older than 1980. So I did and those amounts are what my normal rates are based upon.
The bad thing is that you have to play THEIR game in the first place. But as always, "money" is the real issue plus getting you on the books as the current, registered owner. AND . . . in order to get the tags, you FIRST have to have insurance on the vehicle and then you can get the state safety inspection done. For "normal" vehicle registrations. If this is going to be your ONLY vehicle, it would not qualify for "Antique Plates", which have their own unique set of usage criteria.
There is a LOT of free information on registrations and such on the Texas DMV website. I would hope that other states might be similar.