I saw a definition of a local property of rings in the Stacks project: Link.
In the second line, do they mean that $P(R)$ is true if $P(R_{f_i})$ for some $f_i$ or that it is true if $P(R_{f_i})$ for ALL $f_i$?
My second question is: I have seen a different definition of a local property in terms of localizations at prime ideals: A property is local if it is true for the ring iff it is true at all localizations at prime ideals.
Are these two definitions equivalent? If not, is there some class of rings or which they are?