People in England know lots of meanings for the word cool. As a personality trait, it can mean aloof, trendy, laid-back, sound, or accomplished, and probably many more things too, even before you start thinking of its meaning more widely. Each of these meanings can be quite distinct from the others.
If someone from the UK was asking you what you meant, they were either seeking clarification or being contrary.
Maybe you could choose a meaning from the above words or, -- to be facetious for a moment, -- find someone more co-operative to talk with?
Examples of these different usages:
- Doris has always been warm and giving, but Eva is often a little cool. [aloof]
- Is this where the cool kids go? [trendy]
- No, you're fine. He is cool with high-test peroxide. [unanxious]
- Don't worry about him deceiving you, he's cool. [sound]
- He is one cool accordion player. [accomplished]