Interesting that there is a fine difference between humanitarian aid and humanitarian assistance. I doubt many journalist are this strict about their wording, but good to know nonetheless.
Social welfare provides the exact same things (housing, food, medicine) but instead of some disaster zone somewhere it’s for disadvantaged citizens in your own country. Of course, the logistics are massively different, as well as the timeframe during which the services are provided. But it’s still basically the same thing: help other humans with fundamental necessities.
So I would disagree with your claim that it “isn’t the same thing at all”. It differs in some details, but the central service and fundamental reasoning is the same.
The election system distorts the vote quite dramatically in the US, that’s true.
However, depending on where you’re from left has quite a different meaning from what Americans mean when they say left. E.g. the Democrats don’t want to dismantle capitalism or expropriate real estate companies (the city of Berlin seriously looked into this option). Center left has a similar shift in meaning in other countries.