In the past, it was always called a mosque. St John Philby, and all those types of Arabists, often called it the mosque of Omar.
People hostile to the presence of the Dome of the Rock often call for "that mosque" to be taken down. One doesn't hear any calls for the destruction of "that shrine".
We find, in Crossovers: Anti-Zionism and Anti-Semitism by Shlomo Sharan, Dāwid Bûqay: "Abd al-Malik ... in 691 he built the mosque called the Dome of the Rock" (p 143)
And it has been called a mosque for centuries:
There is an often quoted statement of Muslim historian al-Muqaddasi on the reason for the building of Dome of the Rock. ...
O my little son, thou has no understanding. Verily al-Walid was right, and he was prompted to a worthy work. For he beheld Syria to be a country that had long been occupied by the Christians, and he noted there are beautiful churches still belonging to them ... So he sought to build for the Muslims a mosque that should be unique and a wonder to the world.
I have a 1979 Encyclopedia Britannica on my shelf. I'll quote from their entry on the Dome of the Rock:
The Dome of the Rock was built between AD 685 and 691 by the caliph 'Abd al-Malik ibn Marwan, not aas a mosque for public worship but as a mashhad, a shrine for pilgrims.ML/NJ