Winnie – It is possible that the menorah of Moshe was destroyed/taken by Churban Shilo, like I mentioned before, and the “Menorah shel Moshe”mentioned in shas as in the bais rishon means the “real”menorah, which was lit with a chiyuv (and possibly a bracha) once/twice a day.
About the menorahs in bais sheni, there were three menorahs from the time of the chashmonaim, as we see from the gemara in Rosh haShana (I don’t have a gemara with me, but I’m pretty sure it was on :כד; I’ll try to quote it but I might be off a bit) that they first made it from Shpudim shel barzel v’chipum bbaatz (sticks of iron coated/covered with tin, which sounds like it was probably straight diagonal sticks), heashiru asau shel kesef, chazar vheeshiru asauhu shel zahav.
However, it bears keeping in mind that nothing at all about the menorah is meakev other than the number of neiros (and the amount of oil in them) unless it is made out of gold.
There may have also been another menorah made by Talmai, take a look at Josefus (right before the maisah of Chanukah).