Learn English – ‘the eldest son’

To take it from from a linguistic boffin's standpoint, I guess that "older" is a deictic word: its meaning varies depending on usage. There's a handy illustration at Wikipedia's page for deixis. When you say "my older brother" you put yourself in the deictic centre. Your listener then should figure out the position of your brother relative to that centre.

As WS2 rightfully pointed out, you may with a degree of certainty say "he's my older brother" to an acquaintance who knows that you're the oldest of the flock, but not to just any John Doe.

That's because deictic words require contextual information to be fully understood, and your acquaintance will know that while you're at the deictic centre there's no way for you brother to be at the.. er.. deictic "above", because there's a.. hm.. "deictic ceiling" of the fact of your being the oldest.

Your aunt's children are your first cousins. Their children are your first cousins, once removed. Your cousins' spouses are your cousins-in-law or your cousins by marriage. The people I know generally dispense with such technical terms when they talk about their extended family; they just use cousins to describe the various types.

Your relationship to your relations is not governed by familial nomenclature. Notice that

You'll always remain part of my family

contains no terms of kinship.