Rarity is about how common items are, and only roughly about power
The DMG introduction to magic items states (p. 135):
Each magic item has a rarity: common, uncommon,
rare, very rare, or legendary. Common magic items,
such as a potion of healing, are the most plentiful. Some
legendary items, such as the apparatus of Kwalish, are
By this, and by the naming, it is clear that the primary function of item rarity is to determine how common an item is in the game world. You don't need external designer commentary to understand this. It is right there in the rules. The system is called "item rarity" and not "item power", and its ranks are "common, uncommon, rare, very rare" not "lesser, minor, major, epic". The designers could have just as well ordered items by power, and then commented that the weaker ones are typically more common, but they did not.
Adventure modules do not recommend how common items are, but the core rules do: this is what the magic item tables for randomly selected items in the Treasure section are for (p. 137ff DMG). They mechanically lead to uncommon items being found more often and earlier than rare ones, independent of their power.
From the game world logic, a ring that protects you from cold environments would be much more useful and should be more common than a ring that protects you from, say, Force effects, because cold environments are common, but Force effects are exceedingly rare.
Your question tacitly assumes that a item that is more rare also must be more powerful. This is just not the generally the case1. The rules discuss the power aspect of rarity, and make it clear that there is only a rough correlation between power and rarity:
Rarity provides a rough measure of an item's power
relative to other magic items.
You look at just cold resistance here, but there is no entry for "Ring of Cold Resistance" in the rules. There only is one for "Ring of Resistance". Because of game world logic, many kinds of resistance would be very unusual, for example resistance to Thunder, Radiant, Psychic or Force damage, and you would not want them to appear frequently. The design decision to put all energy types under the same "Ring of Resistance" effectively forces the designers to put it at a higher rarity. Not for power reasons but for how rare many of them would be in the game world.
From a power level perspective a ring that provides resistance and requries attunement is probably correctly placed at uncommon. Attunement is a considerable downside, as it will have to compete with all other such items for your only three slots. Based on the types of damage monsters and spells deal, fire is probably the most useful, although poison and cold also could be of value.
From adventure modules, the ring of warmth shows up quite a bit: in Curse of Strahd, Storm Lord's Wrath, Journeys through the Radiant Citadel, Lost Laboratory of Kwalish, Icewind Dale, Dragon Heist, at least 6 times.
Rings of Resistance however match that: Rise of Tiamat [poison], Storm King's Thunder [cold] and [fire], Tales of the Yawning Portal x2 [fire], Shadow of the Dragon Queen [fire], Candlekeep Mysteries [acid], Dragon Heist [force]: at least 8 times. You also can see that the more useful ones -- fire, poison, and cold -- show up three times as often as the exotic ones, fire alone makes half of the showings. Based on this, a Ring of Fire Resistance would be about as uncommon as a Ring of Warmth, but a Ring of Radiant Resistance would be very rare -- but because all energy types share the same entry in the rules, there is no way to express that in item rarity.
Based on the publication record, neither type of ring is more common -- but beyond what you could earn from selling an item, rarity really has little meaning for items that are placed in an adventure explicitly. Rarity is meaningful primarily if you roll which items show up randomly.
1 And there are clearly many cases where uncommon items can be far more powerful than rare ones, the best known example is the broom of flying vs something like a potion of flying. Or cases where the same funtionality is at different levels of rarity, such as a cloak of protection at uncommon and a ring of protection at rare, even though both confer the same benefit. (As @Yakk observes, different items stack, and this setup lowers the chances of multiple such items being found snd stacking).