Having looked this up in a couple dictionaries now, the person you're responding to is absolutely correct in their objection.
The word "premise" is misused in "on premise" speaking of computers running at a owner's location. I don't think the distinction of it referring to a literal location or metaphorical location (managed by an owner at a different location) matters.
While this might change over time and with on-going vernacular usage, but as of now: our industry is misusing the word. Including myself (until now).
While the person I'm responding to may be upset that 'premise' seems to be acquiring a second meaning in this specific circumstance, they are almost certainly not correct that OP 'want[s]' to say premises. OP almost certainly said exactly what they intended to say.
The use of 'on-premise' rather than 'on-premises' even in formal texts is at least 200 years old (examples below). How much time should it take for us to accept "on-premise", rather than "on-premises"?
The word "premise" is misused in "on premise" speaking of computers running at a owner's location. I don't think the distinction of it referring to a literal location or metaphorical location (managed by an owner at a different location) matters.
While this might change over time and with on-going vernacular usage, but as of now: our industry is misusing the word. Including myself (until now).