I'm quite confused by the terminology *minimal resolution* and *minimal model*.

Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.

Let $E$ be an exceptional component.

Do we have that the intersection number $-(E,E)\geq 2$?

By Castelnuovo' criterion, this has to do with the minimality of $X$. I know one could blow down all the curves such that $-(E,E) = 1$ to obtain a minimal model, but if $Y$ is already regular and contains such curves, then so will $X$ right?

So *minimal resolution* and *minimal model* aren't always the same, right?