Premise has many reasonably common, but diverse senses. Accordingly, it is generally recommended that you just learn the sense (or senses) that currently interest you.
The spelling premiss is less common today, but often seen when discussing a logical premise. (At one time, many insisted this was the correct spelling due to its etymology, but the etymologically different words are pronounced the same and premise has become much more common in usage.)
Her logic is fine except that it assumes a false premise.
Her argument rests on two premises.
I pulled at his sleeve, and we were followed up the sidewalk by a philippic on our family’s moral degeneration, the major premise of which was that half the Finches were in the asylum anyway, but if our mother were living we would not have come to such a state.