Am I technically not Jewish?

So I recently learned that technically to be Jewish you have to have Judaism through your matrilineal line. Now obviously, if you have Judaism on your paternal line, you’re ethnically Jewish, but you’re not “Jewish” according to the laws of Judaism. Well my dad’s side is all Jewish, we have a holocaust escape story and everything. However, on my mom’s side, she grew up without a religion because her mother is Irish Catholic and her dad’s side was Jewish. So technically though my matrilineal line, I have no Judaism. The final twist however is that my mom converted to Judaism when she married my dad, which I guess does make me fully Jewish? But I was curious if I would be considered Jewish if my mother hadn’t converted or if I even am now. For context I went to Hebrew school growing up and had a bar mitzvah and everything, but I’m not sure if that matters.