Both are true, kind of. The gyroscopic stability of the bullet is trying to keep the bullet in the same orientation as it was when it left the barrel but, the aerodynamic force of the relative wind is trying to get the bullet to "point" into the wind. Theoretically, you could spin a bullet fast enough that the gyroscopic stability would overcome the aerodynamic force but you'd run into material limits on the structure of the bullet and there would be a bunch of drag generated.
There's been a bunch written about Spindrift and coning theory that touches on this subject but it gets complicated quick if your not a math whiz.