Sure. "I've eaten already" and "I ate already" mean the same thing.
Right. They are both naturally used in the same way.
Is "I ate already" structurally different than "I ate," "I ate this morning," or "I ate something"?
Those are very different utterances, so I'm entirely not sure what you mean. Anyway, the structure is not relevant here—this is about usage. The quite-perfectly-correct-and-natural
I ate already lacks
perfect aspect. The perfect aspect (shown in
I've eaten already) is what links the past action with the present moment. The utterance
I ate this morning does not need such a link, as it's a simple statement of past events.
What we normally teach (and I really don't feel I should have to be justifying this, especially to retired EFL teachers) is that the word
already, (along with other typical time expressions such as
yet/still/only just, etc.) shows that the speaker has such a link in mind. When there is such a link, the present perfect is the preferred choice, as it shows the link in its aspect.
I ate already, lacking such aspect, does not do that.
Of course, although I do understand that omitting the aspect in
I ate already sounds fine for many AmE speakers, I would question whether such an omission in other utterances also does.
Imagine Karen Carpenter, for instance, crooning "We only just be-gaaaan ..."