D&D 5E Differentiating Arcane and Divine Magic.

[Celebrim]

According to these definitions:

Is a summoner who summons an elemental to do elemental effects, a divine caster?

Good question, but no, because you are the one that did the summoning. The power to bring the elemental was within yourself.

The divine version of this is you make a call to your patron and that patron sends a servant to aid you (or perhaps you get the equivalent of a voucher or a prepaid calling card). The power to bring the elemental came from something other than you.

Quite often an arcane summoner is in danger doing summoning because the summoned being is often unwilling. In the case of being able to summon an elemental, if you are a divine caster that elemental was sent by something with authority over it or at the least it respects you as a fellow ally. In the case of an arcane caster, you are asserting your authority over it directly.