"Seamaster Genève" a separate line or not?

If I remember correctly, the Genève line was introduced in 1953. At that time, it was second only to the Constellation line in Omega's stable of dress watches. From 1965 to 1967, Omega sold watches with "Seamaster Genève" on the dial. After that they went back to "Genève," the line moved more toward an entry level watch in Omega's hierarchy, and the line was ultimately discontinued in 1979. So I'm not sure that the "Seamaster Genève" was a "true" Seamaster, but the early Genèves, particularly from the 1950s, are very nice watches and many have desirable features like arrowhead indices, seconds subdials, crosshair dials, rose gold cases, etc. Very collectible in my opinion..

Spot on. Geneve means both superior dress watch (pre ~1965) and also entry level hack about piece (post ~’65). It’s not that clear cut though, in the ‘60s there were overlap pieces. The Geneve name as a sub-brand went down market once the watches were no longer assembled in Geneva, Omega closed their factory there in the mid ‘60s.

ps of the models shown above, the third, the 041 would be certainly considered entry level.

Nope. For example, there were 135.011's that came with Genève on the dial and the hippocampus on the caseback.