You are probably fine with 2, but I would go for 3

Agree with the above, I would use 1x2 around the perimeter to make it easier since it’ll be minimal amount of cutting. I also don’t like the cross brace of the brackets. The furring strips you linked to will work. However, take a look at the non-furring 1x2s. They are a few bucks more but will be straighter.

Couple of other notes:

1. Pre drill the 1x2
2. I like torx head screws. Grk makes some great products. The R4 or finish screws are awesome.
3. Make sure you screw the plywood into the cleats