Usually if I speak to a fellow dancer I'll just say Lindy, but if it's someone uninitiated it'll be Lindy Hop. And if I'm in a public space where lots of people are dancing, I'll ask someone that's just tapping along to the music "Do you dance?" and not specify which dance.
If things go well from there, I'll ask mid-dance "How's your Balboa/Shag/Blues/etc?".
To me Lindy is just dancing.
The one that always bugs me a little is people calling it "The Lindy hop."
It makes me cringe, and I think it bugs me because it seems like such a ballroom thing to put "the" before the name of a dance.
It makes me think of some demo video from the mid 90's with a super enthusiastic, middle aged couple showing you all the dances you can learn at their studio: "the tango!" "the foxtrot!" "the salsa!" "the swing!" It's a culture thing more than a question of what's correct.
Norma Miller called it "the lindy hop" but if I heard someone call it that today I'd assume they were a very out-of-touch ballroom dancer.
As for the teason: I've heard that LA swing dancers used to call the dance they were doing "Lindy" and not "Lindy Hop" because they had taken the bounce out of the steps.
A smooth swing dancer in LA was dancing Lindy and someone that had more pulse or bounce to their step was doing Lindy Hop. Although this could be a regional difference that never took off, which means that most people wont associate any difference between the two names.
Anyway, using "Lindy" as a standalone word often makes me think of those days, but I don't mind people using it as shorthand for "lindy hop" if they're already on the topic of vintage swing dances.