Mass is a measure of matter. They’re not completely synonymous (you wouldn’t say “I’m made out of mass”, or “my matter is 80 kg”), but they’re close enough concepts that swapping the words won’t lead to any confusion.
As for the c^2, we use a system of units where we have different units for space and time (meters and seconds, for instance). If we used the same units for both, then c would just equal 1, and so we could just say that E = m (or for a moving object, E^2 = m^2 + p^2).
But (typically) we don’t use those units. And so energy and mass have different units, too. Specifically, energy has units of mass times speed squared. So, in such a system of units, it would be not only wrong but nonsensical to say that “E = mc”, or “E = mc^3”, or whatever.