Our knowledge of the inscriptions on the West Door of Moria and on Balin's tomb comes from that part of The Red Book written by Frodo several years after he saw them.
A lot had happened to Frodo during the time between his visit to Moria and his drawing and writing about it. His body, his mind and his very soul had been damaged, so it's not suprising that his memory was a bit vague. He had no camera to record the details.
He could remember what the inscriptions meant but perhaps not the exact words. He used the name "Moria" because that was the name he was most familiar with.
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