> The performance of a classifier can be anything from how fast it runs, to its F1 classification score.

but classification performance would be the efficiency of a classifier algorithm right ? i.e. can i get better classifications / generalizations etc. in lesser time.

Yes and no. It can be better in less time, it can also be just better or just faster (but worse) all depending on what is important.

Classifier A gets F1 score of 0.93 on test set has a lower performance than classifier B, which gets an F1 score of 0.94, regardless of time.

However, you can also say that classifier C, which only achieves an F1 score of 0.929 has better performance, since it is four times as fast.