Thank you all for your answers, especially Fredrick (only thing, you have a typo in your definition of Lorentz transformation: I think you mean g(\Gamma x, \Gamma x)=g(x,x)). I have an additional question: the only definition I know of Lorentz transformations is as linear transformations in...
We are using different languages here. I was thinking of spacetime simply as a linear space and it is in this context that the terminology active vs passive transformations arises. Also, I was secretly thinking of a vector as its components, which is nonsense in your differential geometry...
Physics books rarely make the distinction between active or passive Lorentz transformations. The usual Lorentz transformations of the spacetime coordinates in two different inertial frames seem to me to be passive transformations, because by definition passive transformations are coordinates...