According to the 2021 manual at Mazda USA, some trims have the "TCS OFF" button, others have the off-road button with an icon showing a vehicle climbing a bumpy road. Evidently none have both. Note that the US manual says "some models" for each of the two functions.
This may vary If you're not in the US
Now, prior to introducing the explicit off-road function, Mazda stated there was a stealth light off-roading "easter egg" that could be activated by turning TCS off:
The net effect of the CX-5’s clever i-Activ AWD is that, like the rest of the vehicle, it helps the CX-5 punch far above its weight.
insidemazda.mazdausa.com
One presumes that's still true for trims so equipped. But what's to make of the newer explicit off-road function? One presumes it's some enhanced off-road function compared to the so-called TCF Off easter egg. In any case, I would expect that activating the off-road mode if so equipped turns off traction control, but it beats me what else it might do that would defeat what you're trying to accomplish such as the better fuel economy some claim.