This is what I was going to say.I think half-elf is just a term to mean anything with enough elf qualities that its not entirely human from a human perspective. So this term would cover 1/8th elves, or 1/16th elves and so on until know one would know to suspect. From an elvish point of view one would be half human, right? I think the main. Reason this is not addressed and elves are not typically shown as mating a lot wi to humans and so it is such a rare thing the average human does not encounter them. In a world of your making you can certainly make it common enough that it would be something that was more thoroughly addressed.
I don't typically include elves in my own stories but if I had a situation like this I'd be using "half-elf" too. Trying to intricately explain just how my characters lineage is broken up sounds tedious to write, let alone read, especially if it isn't relevant to the story.
Just call them "half-breed" or your local equivalent and be done with it. Simplicity is a virtue after all.