Putting on the grammar police hat for a minute, there is sometimes misunderstandings about the differences of premise vs. premises. Some definitions for clarity:
noun: a previous statement or proposition from which another is inferred or follows as a conclusion.
"If the premise is true, then the conclusion must be true"
verb: base an argument, theory, or undertaking on.
"The reforms were premised on our findings"
noun: a house or building, together with its land and outbuildings, occupied by a business or considered in an official context.
"Business premises", “He was asked to leave the premises”
Therefore, ‘on premise’ would be ‘on’ a theory, proposition, or argument. To be ‘on premises’ is to be located at the building or structure, which is where software on premises products are installed; in a datacenter, or on physical servers located at a place of business.
There you go, the clarity that ‘on premises’ is what people are referring to when discussing applications installed on servers that the business owns and maintains. While some people think ‘on premise’ is that statement, it is not. And now you know and knowing is half the battle.