Actually it's the plywood manufacturers who recommend a minimum 1 1/4" exterior grade plywood floor (prefer 1 1/2"). Also this figure is intended for the direct installation of tile over wood.
The Tile Council of America and the Ceramic Tile Education Foundation endorses both HardiBacker and Wonderboard products when the manufacturer's installation instructions are strictly followed. Both manufacturers recommend a 5/8" minimum plywood subfloor installed over 16" o.c. joists. Also, screws would be preferable over nails. Because the plywood subfloor in question is 3/4" thick it exceeds the manufacturer's recommendations.
However, 1/4" Wonderboard has a flexural strength of 1500 psi. 1/4" HardiBacker is stronger with a flexural strength of 2100 psi. This higher rating will help to reduce deflection in the subfloor. This is why I recommend HardiBacker for thin panel installations.
TIP - It's simple to determine whether sheet vinyl is perimeter glued- use a toilet plunger to pull suction on the vinyl floor. If the vinyl is loose the suction will give it away. Spot check floor randomly until you are satisfied that the vinyl is firmly attached.