This answer is aimed to be useful for a student learning Latin, not accurate as a historical description of where the conjugations come from.
In my opinion it is best that beginning students use a system that is practical and makes sense to them; historical linguistic notes can be added later unless that is the very goal of studing Latin.
Latin verbs can be (mostly) classified in four groups called conjugations, and the verbs within each conjugation behave (almost) identically to each other.
The four conjugations are distinguished by a vowel:
- Stem ends in ā.
- Stem ends in ē.
- Stem ends in ĭ.
- Some verbs have ĭ is a part of their stem.
- Some put it after the stem (which ends in something else) if a vowel is needed.
- Stem ends in ī.
These vowels make decent labels for the four conjugations.
The order is irrelevant, but I am presenting them here using the standard numbering.
Good labels within the third conjugation are a little harder to come by.
Calling the two classes "vowel stem" and "consonant stem" doesn't quite work, as there are verbs like ruere whose stem ends in a vocalic u.
If you use a numbering, then "III-o" and "III-io" make decent labels.
Or you can say that the stems end in ĭ and ∅.
One option is not to use labels at all and just remember that there are two types of verbs with very but not exactly similar endings.
The labeling vowel is most clearly visible in the active present nominative second person singular.
The first person confuses I with III-o and messes vowel lengths.
The infinitive has a different vowel in III.
You may also choose to label the third conjugation by ĕ instead of ĭ.
One good option is to simply label the conjugations by example verbs.
Speaking of amare-type and audere-verbs and such will make communication easy.
This approach has the benefit that no unnecessary abstract structure is added to the mix, as some students tend to be confused by that.
There are situations where the vowel gets shorter, but the short one of the third conjugation can never get long.
In the end, labeling with numbers works well too.
The labels might be arbitrary but they are simple and common, so they are easily worth the trouble of learning.