An alibi was first a place, not an excuse. In Latin, alibi was an adverb meaning "elsewhere." It was built from alius, "other," plus the ending -bi, which marks where something is.
English borrowed it for the courtroom in the 1700s. Merriam-Webster dates the legal noun to 1731, and Etymonline dates it to 1743. Both dates are for the noun: the name of the plea that a defendant was "elsewhere" when the crime took place.
That legal sense is precise. Etymonline defines an alibi as "a plea of having been elsewhere when an action took place." Wikipedia calls it a suspect's statement that they were in a different place when the offence was committed. It is a claim of complete innocence. You could not have done it, because you were not there. Many other defences admit the act and argue about the circumstances.
The looser meaning, any excuse at all, is much younger. Etymonline dates it to 1912. Merriam-Webster records the verb "to alibi" from 1909. In the excuse sense, being somewhere else no longer comes into it.
Etymonline still draws the old line. Technically, it says, an alibi needs proof of being elsewhere. Anything short of that is an excuse, not an alibi.
The word has a large family. Latin alius goes back to a Proto-Indo-European root, al-, meaning "beyond." The same root gives English alias, alien, alter, alternative and altruism. Etymonline also connects it to the native English else. So "elsewhere," the plain English translation of alibi, is a distant cousin of the word it translates.
Alias makes the neatest pair. Latin alias meant "at another time," and English used it from the mid-1400s to mean "otherwise called." By about 1600 it was a noun for an assumed name. Alibi puts you in another place, and alias gives you another name. Crime writers have leaned on both ever since.