Grateful for help identifying seamaster (if genuine)

Recently I was given my late grandfather's watch and i'm trying to find out whether I should insure it or not worry! It (appears to be) an old Omega Seamaster but there are some things about it that make it hard to be sure or to place:

There is no 'seamaster' on the dial
The marking on the back is a stamped 'SEAMASTER' and 'WATERPROOF' but nothing else

It is automatic and still works beautifully which is great. The glass is pretty scratched so I also wondered about replacing this but the cost could be a lot if this is genuine. Some pictures attached - sorry I din't have tools or skill (or inclination) to open it up!!

Thanks very much for this, it's very much appreciated. I think I shall keep wearing it without worrying too much and maybe service it anyway in a few years as i'd like to keep it going. I noticed (but can't see via camera) there is something like 'swiss made' at the very bottom of the dial but I have a feeling it's got new/different glass on top which is obscuring it -- certainly suggests that along with no 'seamaster' on the dial, it's probably been repaired at some point.

The case reference number is on the inside back. If you get it. You can do a search and it will show you what the dial should look like