Defining 'existence' by using 'interaction' (or worse yet the possibility of interaction) seems to me to be trying to define something fundamental by using something non-fundamental.

I'm not claiming they're ultimate definitions --after all any definition must be grounded in something else-- but at least they disambiguate which meaning is meant, the way “acoustic wave” and “auditory sensation” disambiguate “sound” in the tree-in-a-forest problem. For a real-world example of such a confusion, see this, where people were talking at cross-purposes because by “no explanation exists for X” one meant ‘no explanation for X exists written down anywhere’ and another meant ‘no explanation for X exists in the space of all possible strings’.

As for "mathematical possibility", that's generally not what most people mean by existence -- unless Tegmark IV is proven or assumed to be true, I don't think we can therefore taboo it in this manner...

Sentences such as “there exist infinitely many prime numbers” don't sound that unusual to me.