I first heard the term "snapper" at least 20 years ago, during a peak in the blackjack tournaments offered in Las Vegas. The term may be much older than that.

As to the oldest $2.50 chip, the one you show looks pretty old! Bone? or Wood?

For licensed casinos, it would be sometime after gaming legalized in 1931. There's a 1940's era $2.50 chip listed for Harold's Club in Reno. Also, Jessie Beck's Riverside Casino has a 1959 snapper listed. I've seen the Riverside chip, but not the Harold's Club.

As it was explained to me, the house had 2 motivations for adding snappers to their $5 & up blackjack tables -- (1) it's a time saver in paying a $5 blackjack with a $7.50 payout, and (2) the winning player is more likely to put the entire $2.50 chip back on the table as a bet, rather than tipping the dealer the 50c coin/chip, or pocketing the coin. So it improved the table hold a bit.

Still the practice of using snappers was mixed from casino to casino.

I haven't looked through TCR to find the earliest uses. There are other early snappers in there, I'm sure.
If you're searching through TCR, look for early casinos with a sufficient number of BJ tables to make it worth their while.