forward to the Middle Ages (the Dark Age)

In the European Middle Ages (let's say after Augustin of Hippo) scholars believed what was written by authoritative authors.
So the earth was created some five thousand years ago in six days -- including plants, birds, fish, animals, and snakes -- and Adam & Eve.
And in Jerusalem, there was first The First Temple and then The Second Temple.
After the advent of Modernity, scholars made observations, experiments, and excavations.
And no traces of a central temple for the people of Israel was found in Jerusalem, and what is more: There was no United Kingdom.
IF there were David and Salomon, they were petty chieftains.
Of course, if you are looking for a map of the Garden of Eden in the Hebrew Bible, you can believe the existence of the First Temple.
Or you can read the books of Israel Finkel‎stein (and Neil Asher Silber‎man) and know what can be known.
Even if there are no traces, one can assume that there was a local temple in Jerusalem and that it was destroyed by Pharaoh Šošeq's army.
The Judean King Jehoaš had a second temple built after 835.
Destroyed around 700 BCE by the Assyrian King Sîn-aḫḫe-eriba.
The third temple was destroyed 597 by the Babylonians, and 586 annihilated.
The fourth ‒ the first of regional importance, but still not THE only temple for יהוה ‒ was built by Zerubbabel and Joshua 520 BCE
or 445 by Nehemiah and Ezra (or was that the fifth? )
500 years later Herod the Great built the Sixth Temple.
Around it, he had erected an artificial mount supported by huge stones blocks.
To call this temple the Second is like saying that Pei erected the Louvre and Forster the Reichstag.
And to speak of "Solomon's Temple" is like saying King Arthur defended the UK.
If you say: But the Jews ... I call to your attention that the national socia‎list Ben Gurion proclaimed the Third Reich (בית השלישי) after "his" army had conquered the Sinai Pen‎insula.

CC (in the "the World of the Quran" module) talks of Salomon's Temple, showing that they just reproduce what others say, re‎producing phrases used by Believers in the Hebrew Bible, showing that they live in the Middle Ages, are not "Wissenschaftler".
To say "around the turn of the eras", "in the third century before ..." is better than "in the so-called Second Temple Period", but it is okay. To calls Herod's temple "The Second Temple" is wrong -- and whoever does it is stupid.