Not completely different things, the majority of books all major Christian churches agree on (all of the NT and the majority of the OT). The dispute is really about different views on the boundaries of the same thing - the Christian Bible - as opposed to distinct things
> Just like the word "Deuterocanon" means different things to Catholics vs Eastern Orthodox.
Again, not two completely different things - Catholics and Eastern Orthodox agree on the majority of the Deuterocanon, there are only a few books which Eastern Orthodox have which Catholics lack (3 Maccabees, Psalm 151, Esdras, Prayer of Manasseh; Georgian Orthodox have 4 Maccabees too). And, the Catholic Church has never condemned the Orthodox for accepting these extra books - the Council of Trent condemned Protestants for accepting less, but (quite intentionally) did not condemn the Orthodox for accepting more. So one might say that their status in Catholicism is ambiguous.
> The two English translations provided by the API are not approved by the Catholic church.
I’m not sure how significant that is given that there are a number of Bible translations which do include these books yet which haven’t been approved by the Catholic Church; including one of the translations this site does have (the KJV) - this site’s copy of the KJV is incomplete
> When I search Google for "Bible", the top two results, bible.com and biblegateway also don't include those additional books.
You are mistaken, they both have them:
https://www.biblegateway.com/passage/?search=2%20Maccabees%2...