Minced always means very very finely chopped. That's a mince; it's a preparation technique, not necessarily just for garlic.

Crushing, at least in the context of garlic, generally means using a garlic press (most common in western cooking) or using a mortar and pestle. You can also crush with the side of a knife, bottom of a glass, etc but that's a pretty rare method because you'll usually end up with large globs or chunks and it's very likely that people will get big ass mouthfuls of garlic, which most people don't like (I do like, but most don't)

I can't think of a time when I've seen a serious recipe call for crushing garlic in an uncontrolled manner like with the side of a knife. If you find a recipe that does so, just assume they mean to squeeze it in a press, unless in your judgement the recipe benefits from big clumps of garlic (mashed potatoes is a good example imo).

But yeah, it's either mince or press, if the recipe is unclear. I usually just press no matter what because I love garlic flavor and it's easier than a mince - not because the cutting is hard necessarily, but because it's extra cleaning of the knife, your hands, and cutting board takes more time. Easier to just rinse the press and toss it in the dishwasher.

Slicing is different, as is roasting the cloves.