Do you instantly know what it is or do you have to actively concentrate to the 7th?
one thing to remember with 'diminished' chords is, the way they work is upside down. once you get that it starts to make more sense. so when you say the '7th', it actually doesn't really feel or sound like a 7th of anything. let me give you some examples... key of C. first chord C. second chord, just move the bass note up 1, and keep the other notes, so it's C#, E, G. that's C#dim. so it's like C but with the C note moved up. and the chord actually behaves almost just like A7. so a lot of songs may start C, C#dim, when actually it is a very close sister of C, A7. you'll notice this in a lot of songs, especially from the 1930s or 40s. On the subject of the "diminished 7" well that sounds pretty much the same. it's just a continuation of the diminished thing, it's all stacked 3rds, a very dizzy sound. and again to use the previous example, it would be like C7, then if you move the C note up and keep the rest the same that is just like C#dim7. and again, it's like a substitute for A7b9. wants to go to Dmin in both cases, and most often will. C, C#dim, Dmin. .... but the other one you asked about, half-dim, (or some people call it min7b5) again the 7th won't really feel like a 7th of anything, because diminished chords don't really sit on their feet, they hang from above, as i'm trying to explain, so actually generally what i would say is, a half-dim chord is gonna sound like a minor triad with a certain note in the bass. let's say, Fmin with D in the bass. that's what it sounds like. let's go to the key of Cmin, and let's say you have Fmin, G7, Cmin. if you take that chord sequence, and listen closely, and plug the headphones into your smartphone, you will notice secretly there might be a D bass note under that Fmin. that makes it Dmin7b5 (or D half-dim) instead. the names of these chords are kind of misleading and don't particularly feel like good names for them but that's just the names that are used. probably less so nowadays because this kind of literacy has fallen somewhat out of style .... but the example i just mentioned is generally the most common way you will hear those chords used. the main thing to remember here is that diminished chords mostly mostly mostly usually nearly always do NOT stand on their own and are not used on their own. so if you start to think too much about the chord individually it will not seem to make sense. yet it occurs very normally and commonly as part of a sequence. so the best thing you can do is just look at where it occurs in songs, and just become accustomed to those patterns. and then you will recognize the pattern. and when you hear that Dhalf-dim, G7, Cmin, you will instantly know it is either that OR Fmin, G7, Cmin. but you will have to plug in the headphones and listen close to see if that D bass note is there, to know for sure which it is. but you were asking about the difference between Ddim and Dhalf-dim, well i would say that those 2 aren't the ones you wanna tell the difference between, they're not really even that closely related. i'd say it's more about telling the difference between Dhalf-dim and Fmin. and the difference between Ddim and Bb7. and in both cases you would need to turn up the bass to hear the difference but there is not that much difference. the way you would tell the difference between Ddim and Dhalf-dim is just that they are used in different parts of the song. it's kind of like a wrench and a broom. you can usually tell which one is which because they are mostly not even used in the same context. but the biggest thing i can say, again, is just look at the songs and just find out what happens in songs