Thing is, I want to do a masters in the US, and I do not want to keep studying for the Phd after the master.

And, as I have seen, pretty much all universities offer M.Sc programs, and just a few of them offer M.Eng (which is supposed to be focused on working after the master).

Do you guys think choosing an M.Sc and NOT go for the Phd, will be a good idea? So, start working after the masters, I mean.

That very much depends on what career you want. You should look at job openings and see what qualifications are required, and see if they specify MSc, M.Eng or PhD/doctoral equivalent.