You’ve been coughing and your throat feels irritated. For relief, you put a medicated candy-like tablet in your mouth. What do you call it?
Cough drop. I don’t think I’ve ever heard the word lozenge used(despite it’s presence on packaging).
If it’s cough-related, I always call it a cough drop.
I have used some designed only for a sore throat (something like this Cepacol one) and I mostly call those by brand name, but might fall back on lozenge. I certainly wouldn’t call it a cough drop because it is not intended to treat a cough.
Demulcent
Cough drop. If it’s diamond-ish shaped I call it a lozenge, because that’s what a lozenge is, rhombus-shaped.
Cough drops, lemon drops, horehound drops, et al are named drops, because after sheets of said candy are cooled, they are DROPPED on a solid surface, to break them apart from each other or the matrix in which they were formed. Talk about calling a spade a spade. 
Well, I call it a cough drop, but I don’t use it because they upset my stomach.
Cough drop, unless it involves zinc in which case “zinc lozenge” sounds fine. If someone asked me to hand them a generic “lozenge,” I wouldn’t look at them like they have two heads, though.
Really though, the pedant in me says it should only be a lozenge if ◊ shaped.
I call it a Hall’s Mentholyptus.
The only person I’ve known who ever used the word “lozenge” was my ex, who also spoke of “vehicles” and “coitus.” Yes, he sounded like a cross between Sheldon Cooper and Data.
Cough Drop. My grandmother would say lozenge and I thought she meant a lounge chair. English is tough on the young.
My wife always calls it a “lozenger” and I’ve picked that up. She’s a native Arkie, but that doesn’t explain it.
Usually, I just say Hall’s, as in, “My throat hurts. Do we have any Hall’s?” (Hall Mentholyptus, obviously)
If it is any other brand (and really the only other brand we ever have is Luden’s Cherry), I say cough drops.
If it is hard and candy-like in texture (basically smooth to the touch) I call it a cough drop; think the usual Hall’s. If it more a faster dissolving thing like a “Thannis” or “Fisherman’s Friend” with a grittier texture, I call it a lozenge.
Understandable, but keep in mind that the origin of the word is “lausiae”, which means small pebbles.
My mother called it that as well–“lozenger,” with a definite -er sound on the end. She lived all her life in Indiana, but I’ve never heard anyone else from around here say it that way.
I used to think* there was a clear distinction between the two: if it’s medicated, and it’s for a cough, then it’s a cough drop; if it’s basically candy, and it’s for a sore throat, then it’s a lozenge.
Then one day in high school, I had a bad sore throat, and my friend gave me a handful of “lozenges”. I took maybe 6 or 7 of them, one right after another, until I suddenly found myself getting very, very sleepy. “But you said they were lozenges, not cough drops! How was I supposed to know they had cough medicine in them?!”
*And I still think of them this way, although I now realize that this is non-standard.
I’m pretty sure I’d call it a cough lolly. But I would know what the other 2 phrases referred to.
Or a pastille.
I say throat lozenge, as in “do you have a throat lozenge?”.