The Monster Manual defines elementals
Elemental is a Monster Type (MM 6). It is defined as:
Elementals are creatures native to the elemental planes. Some creatures of this type are little more than animate masses of their respective elements, including the creatures simply called elementals. Others have biological forms infused with elemental energy. The races of genies, including djinn and efreet, form the most important civilizations on the elemental planes.
Other elemental creatures include azers, invisible stalkers, and water weirds.
From this description, it is clear that the fire elemental is just one example of what a fire elemental can be. Any creature native to the Plane of Fire is a fire elemental, and would be treated as such for the purposes of the ring.
This understanding is further confirmed by two spells found in the PHB. In conjure minor elementals, we read:
You summon elementals that appear in unoccupied spaces that you can see within range. You choose one the following options for what appears:
One elemental of challenge rating 2 or lower
Two elementals of challenge rating 1 or lower
Four elementals of challenge rating 1/2 or lower
Eight elementals of challenge rating 1/4 or lower.
Note that these creatures are simply called "elementals", and that none of them can be CR5, that of the elementals. Upcasting the spell conjures more of them, but does not increase their power. The spell says that "The DM has the creatures' statistics", so it is the DM who chooses whether they are fire elementals or some other element.
Then we have the conjure elemental spell, which says (emphasis mine):
You call forth an elemental servant. Choose an area of air, earth, fire, or water that fills a 10-foot cube within range. An elemental of challenge rating 5 or lower appropriate to the area you chose appears in an unoccupied space within 10 feet of it. For example, a fire elemental emerges from a bonfire...
Here, you can specifically conjure "a fire elemental" by targeting an area of fire, but note that the elemental conjured is CR5 or lower, meaning it is possible to get the fire elemental, but also explicitly possible to get a fire elemental should the DM decide.
Further, as Nobody the Hobgoblin points out in a comment, the conjure elemental spell can be upcast, and when you do so, "the challenge rating increases by 1 for each slot level above 5th."1. If the fire elemental was the only or the primary creature conjured by this spell, there would be no reason for this ability, whereas, for example, upcasting an air elemental to CR6 might bring you an invisible stalker, as suggested by the additional text on the DDB description of the spell.
As just mentioned, on D&D Beyond2, the spell description goes, well, beyond what is written in the PHB and lists magma mephits, magmin, azer, and salamanders as examples of what you might conjure with this spell in addition to the eponymous fire elemental, further supporting that "a fire elemental" does not mean only the fire elemental.
The Moon Druid's elemental wild shape is an exception
As pointed out by @PeterCordes in a comment, Moon Druids can gain the elemental wild shape ability, which says:
At 10th level, you can expend two uses of Wild Shape at the same time to transform into an air elemental, an earth elemental, a fire elemental, or a water elemental.
Here, we uncharacteristically take "a fire elemental" to mean only the fire elemental. This is clarified by a Sage Advice Compendium ruling which says:
Does the druid’s Elemental Wild Shape limit you to the four creatures listed, or can you turn into any creature with the elemental type? The creatures listed in Elemental Wild Shape—air, earth, fire, and water elementals—are specific creatures in the Monster Manual, not creature types or subtypes. Elemental Wild Shape allows you to transform into one of those creatures.
It should be noted that Sage Advice is saying "this is how elemental wild shape works", and is not saying "this is what 'an elemental' means in rules text". The ruling here is a specific exception to the general rule established the Monster Manual.
There is no explanation given for why this particular ability is more restrictive than, say, the spells to which a Moon Druid already has access (conjure elemental, conjure minor elemental), but it may have to do with scaling the CR of the elemental form. In their wild shape, Moon Druids are permitted to assume forms "with a challenge rating as high as [their] druid level divided by 3, rounded down", and at 10th level this would be CR3 (the fire elemental is CR5). The elemental wild shape ability is separate and distinct from the wild shape ability, though, and has no CR limitation any more than it requires the druid to have previously seen the form in question. However, if elemental wild shape both permitted any elemental form and were untethered from CR, then a 10th level druid could take the shape of, for example, a legendary CR16 phoenix. Thus using the same CR scaling as for wild shape would not permit a druid to have access to the fire elemental at the level they received elemental wild shape, but having no scaling would permit them forms that are clearly over-powered for their level. I think it is likely that designers had to choose between creating a new CR scaling for just this ability, or simply restricting elemental wild shape to the eponymous elementals (which also may have aligned better with previous editions), and they chose the latter.
1 It is not obvious what increasing the challenge rating of a creature of "5 or lower" does. Does it increase the maximum, the minimum, or both? That is, if I cast conjure elementals at 6th level, would I get an elemental with a CR range of 0 to 6, 1 to 5, or 1 to 6 (all inclusive)?
2 Whether this table (or that included with conjure minor elemental) is "official" content is debatable, and see this comment by user10063. What is not debatable is that the PHB in both spells calls creatures of less than CR5 elementals, and in the second spell explicitly says that conjuring fire elementals of less than CR5, and probably more than CR5, is possible.