From my understanding, tempo markings (such as vivace, largo, etc) give the overall direction while metronome markings give the precise speed the composer intended. I was also taught that time signatures are there purely for notational purposes and have no effect on the tempo of the piece.
However, I’m getting lost at some inconsistencies between these 3 concepts. Take, for example, Chopin’s Etude Op. 10 No. 5 (Black Keys). It’s written in 2/4 time with the tempo marking Vivace, and Chopin specifies a M.M. of 116. However, when I listen to pianists perform the piece, they seem to be performing it significantly faster than what is written. Does the piece being in 2/4 mean that the M.M. should be doubled as it’s no longer 4 beats per measure?