Just trying to keep the narrative simple, yes there’s Akkadians in Mesopotamia before then but they’re not really in southern Mesopotamia where Sumerian was spoken. I said ~2300 BCE because it depends on what chronology you use, 2334 - 2279 BCE is the middle chronology dates given for Sargons reign but that can vary my +- a century so I tend to give rough centuries for events. You’re right that saying they migrated from the levant is probably a misnomer and I was just basing that on Akkadian being a semetic language, I’m not as deep into the anthropological or linguistic side of scholarship so migration theories aren’t my strong suit. And I should clarify that I’m saying Akkadian presence because it’s only after the rise of the Akkadian empire that semetic speaking people become a strong presence in southern Mesopotamia, whether that’s a migration of them or locals adopting their cultural practices is still up for debate among scholars.

Amorites are arguably first attested in the Ebla archive from the ED period though that’s northern Mesopotamia, and in southern Mesopotamia Naram-Sin claims to have fought Amorites on one of his campaigns. It isn’t until the Ur III period in the 22nd and 21st century, after the 2300 date I gave, that we have evidence of Amorites integrating into Southern Mesopotamia, and of course the famous wall to repel the Amorites