Agreed 100%.I think you're looking at it the wrong way. Someone not wanting the race (and everything that comes along with it) to be completely ignored doesn't mean they want to be defined by their race. A persons race is just one aspect of who they are, but it also impacts how they interact with society at large.
I don't think anyone would expect you to feel defined by your race as a white male, but that's because you're seen as the default, especially in the United states. For instance, if you were raised in an environment where you were the ONLY white person and you rarely saw someone who looked like you in the media you'd be more aware of your whiteness. But in the U.S. it's not something you even have to acknowledge.
For someone like me though it's something I have to acknowledge at times because I'm not seen as the default. I always use the example that I once won an office raffle, and the prize was voucher for a tanning salon.....I'm a dark skinned black man, what the hell am I supposed to do with a voucher for a tanning salon. The people in the office obviously meant no harm in it, but when coming up with the prize they viewed themselves and their experiences as the default and just assumed it'd be a nice gift for everyone. They ignored the race of every black person in the office and in doing so ignored that what might be cool for them might be utterly useless to me.