Is this 344 Bumper a real Seamaster or not (2807)

Hello folks,
I recently bought this nice Omega 2807 (344) from 1954 from the bay.

The Seamaster wording was already peeling off and it looks clearly applied later.
Then when cleaning the dial, the wording went off very easily.


I was ok with that, thought: ok, not a real Seamaster, no problem. But then I came around these examples:

https://www.ebay.com/itm/274805236076

Clearly applied later. If you're searching, you'll find more.
So my question is: Were there any Seamaster (dials) that were re-branded from Omega during production? Or all these all counterfeits to increase the value?
Does a real Seamaster require the SM wording engraved on the caseback?

I ask because I'm looking for an original SM from '48 or '49.

Just for the record: The final result after restoring:

Another example: (but the S is wrong)

https://www.ebay.com/itm/166265165876

Your example definitely was original and Seamaster was not added later. You should study this thread. Plenty of variations of the S in Seamaster as well so that eBay example is correct.