I have been under tornado watches here (not any warnings since I got solar/PW.) I have not seen it activate storm watch. I believe storm watch does not activate for watches - only warnings. Typically the watches are accompanied by severe thunderstorm watches and/or warnings.
Further, I have seen it not activate for thunderstorm warnings or for flash flood warnings. So, my assumption is it only activates for a subset of warnings. It is possible Tesla feels these warnings either don't get enough notice before the event to do much good, are less likely to create an issue (flash flood may be an example,) or maybe even are too common to use for storm watch (since I am not certain there is anything that would officially authorize its use, particularly for those claiming the ITC, so over-use could conceivably put somebody on the wrong side of the thresholds for claiming it.) It is also possible that I just didn't notice it (though I know I have not gotten notifications of any storm watch events.)
It would be helpful for Tesla to list the rules for storm watch - exactly what NWS events trigger them and if there are any other caveats (like not triggering if the warning is for an hour or less, as an example.)