TL;DR This GM limits summon spells to bringing forth only elementals in the elementals sections of the Bestiary and Bestiary 2. (We're not using the problematic aether elementals from Bestiary 5 until there's clarification.) This GM believes this an accurate and fair ruling based on the pattern established by the summon spells' creature lists, but the game never flat-out denies any alternatives.
The term elemental is both a subtype and a kind of creature—a creature that has in its name the word elemental like the air elementals. While the creature lists on individual summon monster and summon nature's ally spells do not make explicit distinctions between these two different uses of elemental, each list does have a separate second column devoted to an entry's subtype: the entry elemental (Small) in the first column has as its subtype in the second column elemental, for example.
Although it would strike this reader as a strange editorial decision, this reader admits that, alone, the mere presence of the elemental subtype in the second column doesn't necessarily in this case exclude the first column from also meaning the elemental subtype. However, viewing the term elemental as it's used in the wider context of all those summon spells leads this reader to believe that those lists in their first columns mean elemental not as the subtype but, instead, as the kind.
Summon spells and creature lists
As summon spells increase in power at lower levels, so, too, do the sizes of elementals that can be brought forth with them. For example, a summon monster II spell can bring forth an elemental (Small), a IV an elemental (Medium), a V an elemental (Large), and a VI an elemental (Huge).1 Were the summon spells' creature lists' elemental entries to stop there, the reader would be utterly at a loss, the game implying no distinction between subtype and kind.
However, as summon spells' power increases at higher levels, the elemental entries on those later spells' creature lists stop using size and switch to kind. For example, a summon monster VII spell can bring forth an elemental (greater) and an VIII an elemental (elder).
In sum, the elementals that summon spells can bring forth are, in order, Small, Medium, Large, Huge, greater, and elder. Those parenthetical notes in summon spells' creature lists' elemental entries correspond to the kinds of elementals in the descriptions of the traditional Bestiary elementals (air, earth, fire, and water), the Bestiary 2 elementals (ice, lightning, magma, and mud), and the lone Bestiary 5 elemental (aether).
Rather than viewing this as a remarkable coincidence or a happy accident, this reader views this as a pattern. Further, this reader believes that this pattern establishes that summon spells are to be limited to bringing forth only those elementals that also possess kinds corresponding to all of those categories. A convenient pattern like this eliminates confusion, smoothes an otherwise awkward progression, simplifies references, and facilitates play.
This conclusion is subtly reinforced by the idea that were that first-column elemental supposed to indicate subtype rather than kind then the low-level elemental entries that also have size categories would be, in that regard, unique among the summon spells' creature entries. All of the other creature list entries are of a kind of creature rather than a type (a dire wolf rather than dire animal or animal (Medium), or even just animal alone) or a subtype (bearded devil rather than devil (Medium) or just devil alone). Even in the entry mephit (any) the term mephit is a kind of creature named mephit—mephit isn't a subtype, either.
Still, because proving a negative in an exception-based role-playing game system is just that difficult, another reader may hold the opinion that this pattern is coincidental or accidental rather than purposeful and that the actual goal—or, at least, the rules-as-written result—is chaos. That is, to my knowledge, no developer has said definitively that elemental in the first columns of the summon lists doesn't mean subtype.2,3
This reader, though, finds the potential impact of that opinion on actual campaigns too vast to consider.
On the other hand…
Another reader could argue that spells should be read in isolation and without regard to other spells. And, therefore, because spells stand alone, summon spells' creature list entries that are an elemental of a particular size can be viewed as allowing the option to summon any creature that possesses the subtype elemental and that's also that size. And, that reader could argue, by extension it's only the greater and elder elemental entries that are restricted to creatures both that possess the subtype elemental and that are greater or elder for some other reason like by having the words greater or elder in their names.
Make no mistake: this reader agrees with that principle: game elements should stand alone, shouldn't require referencing other elements to realize their meaning or potential, and should make themselves crystal clear through their own words. Unfortunately, while this reader agrees with that principle, he still disagrees with the argument. Context still leads this reader to believe that the perceivable pattern means anarchy shouldn't reign here. (Moreover, were the Pathfinder Core Rulebook as comprehensive as that other reader wants it to be, it wouldn't fit in your backpack or, in all likelihood, even your car!)
Also, a word of caution: A GM that rules that the lower-level summon spells can bring forth any creature that both possesses the subtype elemental and is the appropriate size is probably already aware that this expands the summon spells to other creatures like, for example, the aerial servant.4 However, that GM might be unaware that the ruling could also open the door to a savvy player having his PC use summon spells to bring forth creatures to which have been applied one or more templates… so long as templates both don't make the creature too big or too little and don't remove from the creature the elemental subtype.
See, normally, applying one or more additional templates—beyond what a summon spell may mandate—to creatures brought forth with a summon spell is at least tacitly prohibited by the summon spells' creature lists' first columns limiting creatures by kind. For example, a dire rat isn't merely a dire rat anymore if to it has been applied the template fungal creature; instead, it's a fungal dire rat, and there's no fungal dire rat entry on the summon spells' creature lists. (See the fungal lizardfolk for an example.) Contrast this with a fungal aerial servant, for example, that's still a Medium creature and that retains its elemental subtype (although its type becomes plant, and it gains the subtype augmented (outsider)). Under an expansive ruling of what that first-column elemental means in the summon lists, that fungal aerial servant is a valid creature that's available from the summon monster IV or summon nature's ally IV spell!
With this in mind and even without the addition of third-party templates, the options that become available through just a summon monster II or summon nature's ally II spell multiply in variety and power so as to be very nearly overwhelming. The GM should expect skilled, research-minded players to bring forth crazy powerful creatures with their PCs' 2nd-level spells. That GM had better be prepared to accommodate—or meet head on—PCs capable of bringing forth such creatures lest the PCs ride roughshod over any opposition not similarly reinforced.5
1 Due to a stylistic choice inherited from D&D 3.5, Pathfinder capitalizes size categories like Small, Medium, and Large. The elementals on the summon lists are no exception, exacerbating this confusion.
2 Creative director James Jacobs says few creatures are added to the summon lists (Dec. 2010) because such additions complicate the game (May 2011) and just because that's why (Dec. 2010). A reader may also be interested in Jacobs's explanation for the presence of giants on the summon nature's ally lists (Sep. 2015) and the absence of genies from the summon monster lists (Aug. 2009).
3 In a Paizo messageboard thread mirroring the question, this post (Apr. 2014) mentions Jacobs alluding to the summon spells being limited to creatures from the first Bestiary. This reader humbly suggests that the user may be remembering Jacobs's post (Nov. 2012) that mentions Bestiary 2 elementals being okay for summon spells and elemental body spells.
4 Under such a ruling, a summon monster IV or summon nature's ally IV spell can bring forth a CR 11 aerial servant. Absent that ruling, the same spells could bring forth only, for example, a Medium air elemental. A Medium air elemental is CR 3.
5 Note that here with third-party sources, a level 3 wizard can cast a summon monster II spell to bring forth a cataclysmic void cat, a Small elemental that'd be hilarious were it not, like, CR 30.