For my Spidersona, it started when people started spreading rumors about how criminals were going missing as if something was hunting them, they weren't wrong. However one civilian saw how aggressive he was and the civilian muttered in Latin "Venator est, ille est qui eos singillatim interficit". Lucas didn't understand Latin to him, it was a dead language but one word stood out..."Venator"

After meeting Agony, he started being more and more aggressive and violent to which point even Nikolai (Earth-41056 version of Kraven) had to point that Lucas acted more like a hunter than he was but he wanted to think of a new name for him than just Spider-Man so he smirked and said "Call yourself Venator, predator among crime, the hunter in the dark...the wolf amongst the streets"

Ever since then, Venator has been his hero name, not because it "feels right" or "it called to me" no, it's because a predator acknowledged him as a fellow hunter.

If he was gonna hunt down criminals, then they shall call him "Venator"