Indistinguishability versus semantic security?

I found some foundations of crypto course notes that mention that these two are equivalent statements of their own degrees of security and was given the following definitions:

I was always under the impression that the second definition (the one on the bottom) was the prescribed one for Semantic Security, and that computational indistinguishability was a completely separate thing? I do not understand the necessity or intuition behind the need for a "simulator" as it is called. Can someone shed some light?

• Course notes are meant to be used in a classroom setting, where an instructor is available to fill in such gaps. It sounds like a self-contained textbook would be a more appropriate source for you. – fkraiem Nov 7 '17 at 5:35
• @fkraiem thanks for the advice, you're right that asking about course notes would be better directed at the teacher writing them, but all I have access to is the textbook the notes are based off, Foundations of Crypto, which doesn't offer information on the equivalency of semantic security and indistinguishability. If you have something to add to K.D.'s answer, or an interpretation of your own it would be much appreciated. – z.karl Nov 7 '17 at 20:08

This is formalised by simplifying the problem: the adversary chooses an "interesting" predicate $f$ on the message space and a way to choose messages such that the predicate holds with probability $1/2$. Then we must prove that the adversary cannot determine the value of the predicate with probability significantly different from $1/2$. (You can define a variant of perfect security in exactly the same way.)