Undercooked has a broader meaning.

A steak can be undercooked but never will be al-dente.

Undercooked can mean improperly cooked, al-dente always means properly not cooked too much.

"Undercooked" means "cooked too little", al-dente is not "cooked too little", it's "cooked enough" in order to be al-dente. I guess you could say al-dente is more "not cooked too much" instead. Personally I cannot stand pasta that is not al-dente.

Sibling comments here are good but what I think they are missing is a nuance of artistry that underlies a lot of GP’s examples.

So in this case, al dente means cooked to perfection. Not too much, not too little, and is reflective of the craftsman’s skill.

(I'm not Italian, this is wild speculation) I think it directly translates to something like "to the tooth", so I always imagine it meaning a food having a certain kind of "bite".

It simply means that you feel a degree of resistance or firmness when you bite an al dente piece of food, ie. it has a "tooth feel" unlike soft, fully cooked starchy things.

An engineer might say that al dente has more shear strength left than fully cooked.

Al dente means the pasta "bites back". Its cooked, and its just cooked enough that there is still some bite in it. It's not turned to mush.