It's worth noting that we have games where Mini refers more to size, as well as stuff like the Poison Mushroom that changes it, so the word Mini would be rather confusing as is. He's not a Miniature Link. It's more Minifig Link, of course, but both words apply if you shorten it. And I'm aware Miniature can also be a term to mean a Minifig too, but shortening it is still strange. Minifig Link wouldn't be a bad name, though... though I guess that's why Toy Link is somewhat of a nickname these days. There's also the funny nickname of Christmas Special Link. But either way, I'm sure whatever name it is, it'll be rather apt.
Outside of a bad translation at worst(coughDuckHuntDuocough. For minor context, the trailer notes that there's a 3rd person, being the player, as part of the moveset, but it also has multiple characters cameo, so it's a pretty misleading name. It's not super bad, unlike the poor Pikmin Captains, who are actually outright misnamed, directly ignoring that the Pikmin are part of the character. Instead of a minor error, it's a complete overhaul of what they should be... for some reason. Albeit, Pikmin & Olimar doesn't sound great in English either, but they did a fairly poor job at conveying the meaning outside of Japanese too).