Edit: I just realized you picked some tricky sentences there. It's not a good example of す v.s. せる I meant in my original post. It's true that す is the 文語 (i.e., the language register mainly for writing) version of せる. But there's much more to it when it comes to 〜す v.s. 〜せる.
Apparently, the verb in the latter example sentence you chose is a 五段活用 one because it uses し with its 連用形 to connect it to て (i.e., 空か->"し"->て). This is considered an equivalent, colloquial form of the normal 〜せる version (except that the 五段活用 〜す version can't be used as a honorific language as in あらせられる.). So it's simply a 五段活用 version of a verb in causative form. (If you haven't learned causative yet, read the revised version of my first post below. 〜せる is an auxiliary verb that adds the sense of "causative" with a variety of nuances). Between the two, there isn't that big of a difference in nuance except that the 五段活用 version is more of spoken language and sounds rather direct. Also, I feel it's more common in the Western part of Japan, though I don't have any concrete data to back this up.
I revised the original post below so it can read as a correct answer to your question. I'm sorry for the confusion.
In general, せる is an auxiliary verb that can add various meanings and is typically classified as causative. Here, せる is adding the sense that something is put in an undesirable situation, although if I were you, I'd simply accept the meaning and usage of the whole chunk of (お腹を)空かせる without parsing it into grammatical pieces; I'm a native speaker, and it took me a good minute to figure out what that auxiliary verb is doing there.
So, 空かせる is not the potential form of 空かす. If you really want to learn the various meanings and usages of せる as a tiny grammatical portion in a word, 明鏡国語辞典 seems to be very good at explaining them in this way if you're ok with a monolingual dictionary for native speakers. (And this dictionary does mention the colloquial 五段活用 version in a grammar note in the entry for せる too.)