It can be put that way, but Herod's Temple was basically a rebuild and a continuation of the Second Temple.
No it wasn't. Herod totally destroyed the second temple... removing the foundation stones and grating the bare land.
Why did he do that ?
To understand... one must understand who Herod was and the history,
Herod was NOT an Israelite. He was an Edomite "Jew." In the 2nd century BC, Edom [Israel's enemy] lost a war
against Israel. Then the prince/high priest of Israel did something very evil and stupid. He allowed Israel's enemy,
Edom, to live in the Kingdom of Israel if they "converted"... which they all did en masse. These Edomite fake "Jews"
proceeded to usurp and disinherit the Israelites. In 37BC, Herod with the help of Rome conquered Jerusalem and
Rome proclaimed him, King of Judea. Herod proceeded to murder all the Israelite royal heirs which brought to
an end the last Israelite Hasmonean dynasty and began the Edomite Herod dynasty. So, Herod totally destroying
the second temple and rebuilding his own... was part of his mission to disinherit Israelites by destroying their heritage.
Once one understands how dispossessed Israelites had become in their own land... one can understand why
God chose that time in history to incarnate in His people to save them. Just imagine... at the time of Christ,
the vast majority of Israelites had been scattered into Asia and Europe since the 8th century BC... most of them
now "gentiles"... while the much smaller group of Israelites in Judea had been dispossessed by a pagan empire
[Rome] and Edomite fake "Jews" who had not only disinherited Israelites... but had taken over their identity.
Revelation 2:9 and Revelation 3:9 also refer to these fake "Jews" for the same reason that Paul quoted the OT
saying God loved Jacob [Israel] but hated Esau [Edom] Romans 9:13 .... because that conflict... the confllict
between Israel and Edom that began in the OT... continued in the NT... still continues to this day... and will
continue until Christ returns [Luke 19:27.]
.