I compared these five razors and threads.
The shovelhead and 1912 have the same threads and are interchangeable.
The 1902 Lather Catcher and the ER 1914 have the same threads and are interchangeable.
The silver Gem 1912 has different threads from the other 1912: not even close. They do not interchange.
I found that this #8 sheet metal screw will work on both the 1902 LC and the ER14. I cannot however tell you the official size of these threads. I think they are 8-24.