The Temple Institute calls it the Second Temple, as well as Israeli archaeologists. The reason is that the Tamid Offering was uninterupted except for a break of 1150 days during the time of the Maccabean Revolt.
Yeah, I know that most do; I was just saying that there are some who don’t use that convention. So, the other poster who mentioned that wasn’t just making that one up. It does come from somewhere.
As for myself, I use the convention of the Third Temple coming up (in the future)...