Identify similar Game to Achi
Predominantly games of this style fall under either the Alquerque or Zamma families.

There are a number of variants played globally, with the board shape and number of stones varying considerably depending on region.
Games typically grouped in the Zamma family tend to have larger boards and stone counts, where Alquerque variants tend towards the smaller.
Though the grouping into families for this style of game is relatively arbitrary, and is mostly dependent on which game the individual doing the grouping first came into contact with.

As far as I can ascertain the 3 bead variant was developed by the Gujaratis, but as with a lot of these games there's no clear literature as to when the variant came into favor, or even of it's actual origin.

For similar games, the closest I'm aware of  in terms of board size would be Four Field Kono from Korea, but that has a modified capturing mechanism.
There are a number with almost identical rule-sets, so just searching for Alquerque is probably a good option.