The same two notes are two different problems. Played at the same time — harmonically — they fuse into a single colour, and what you are judging is roughness, width, and blend. Played one after the other — melodically — they become a movement you can sing, and what you are judging is how far your voice had to travel.
Which one to drill first
Melodic first, almost always. A melodic distance can be checked against your own voice, and a self-check that costs no equipment is the fastest feedback loop in ear training. It also matches how most people already store music: as tunes, not as vertical slices. Once the melodic version of a distance is solid, the harmonic version has something to attach to.
Why harmonic is harder
Because fusion is the point of harmony. Two notes sounded together are heard by the auditory system as one event with a texture, and separating that texture back into two pitches is genuinely extra work. The wide, consonant distances survive fusion best; the narrow ones turn into beating and roughness that is easy to notice and hard to name. Expect the harmonic version of a distance to lag its melodic version by a while, and do not treat that gap as failure.
Mixing the two
The useful drill is not one or the other but both, unpredictably, in the same session — because in real music you get no warning about which one is coming. Run a level in melodic mode until it is comfortable, switch it to harmonic, then set it to mixed and accept that your accuracy will drop. It comes back within a few sessions, and what you have afterwards is one skill instead of two.
Direction is a third variable
A melodic distance also has a direction, and the two are not equally easy. Rising and falling versions of the same distance are stored separately by most learners at first, which is why the anchor tunes on every interval page are labelled with the direction they move: a rising fourth and a falling fourth want different reference melodies. Harmonic rounds have no direction to get wrong, which is one of the few ways they are simpler.
Practically: get a distance solid rising, then drill it falling, then let the level mix both. Skipping the middle step is the usual reason a distance feels reliable in practice and collapses in real music.
What this means for chords
A chord is the harmonic problem with more parts, which is exactly why singing the stack back works so well: it converts a vertical, fused event into three or four movements you made yourself. Once you can do that, the quality drill becomes a description rather than a guess.