The distinction between a cutter and a slider is extremely faint; there is no clear line defining when it is one or the other, and not a lot of pitchers actually throw both regularly.  So I think saying that definitively one is one thing and one is the other is hazy.  I would tend to call a cutter a fastball and a slider a breaking ball, but the cutter to a great extent is a little bit of both.  To say that the delivery of one is altered and the other is not seems silly.

I think the velocity is the deciding factor. Most major league fastballs, even four seamers move somewhat. A cut fastball is usually going to be thrown in the high 80s and low 90s, where most breaking pitches are thown in the mid to low 80s.

If Freddy Garcia threw a cutter it would be slower than many pitchers' sliders.  I think absolute velocity is a lot less significant than relative velocity...

I agree. But his four seamer would also be slower most pitchers cut fastballs. No one would consider his slow four seamer a breaking ball.