Migrant probably wasn't always a dirty word. But it's since become that due to how it's used in the media. Someone who moves legally to a country on a work or student visa is still a migrant, but they won't be labelled that way permanently. You wouldn't talk about the café down the road run by migrants just because the owner was from abroad.
So usually if you see that word used in a headline you can assume it will be talking about asylum seekers or "illegal immigrants" or something similar, and it's become almost an insult which helps to dehumanize people by making it their core identity (ie focus solely on one aspect of their lives rather than just treating them like people who happen to be migrating for whatever reason).
Am I special? 🤔 I guess we're all special in our own way. If someone broke into my home, I also probably wouldn't invite them to sit down for dinner, so we're probably the same on that. But people fleeing persecution or conflict is not quite the same as a guy with a crowbar. As far as I'm aware, actual clearly illegal immigration is more likely to be someone overstaying their visa than arriving on a dinghy seeking asylum.