While I believe the character should be allowed to conditionally hide their Distinctive Style when it makes sense (no character is going to go on a night run with an incandescent tattoo giving away their position, nor is any other character going to go on a run with them, lol), but it's definitely true that if the player is going to take the flaw, then the character is going to have to display it whenever appropriate, even if it isn't always advantageous.
Distinctive Style is another one of those qualities that players don't think hard enough about. Having a distinctive style is very dangerous for someone who lives a life that typically prefers anonymity, or at least subtlety.
As far as the OP, the way I read the rules on Distinctive Style is not that it is a variable trait but that it can be taken more than once for more BP. Though it also says the cumulative dice modifiers are capped at +6 so I don't know why a GM would allow a character to take it more than twice. I guess you could have multiple styles that translate across mediums. Like a distinctive Matrix style that wouldn't show off the Net unless you were conversing with other Hackers, or a distinctive way of talking that won't be apparent unless you speak or are audible.
Very poorly written Quality, and it's the longest and most heavily explained of them all, lol. I mean you could have a giant AR billboard that followed you around pointing out who you were and you'd still only get a maximum of +6 to identify you. Seems like the best bet is to keep Distinctive Style NQs in the 5 point range, otherwise you're starting to give away points.