There aren't necessarily any hard differences between a Software QA Engineer and a Tester, but frequently the former title will tend to be a role with professional testing experience.
The most important thing is to check what's on the job spec, and see what is expected of the given role. Many roles advertised as a Tester role will be a fairly standard software testing role without requiring any specific skills in test management or test automation, whereas with SQA Engineer roles there may be some expectation of those skills.
If you're working as a tester and you have a few years experience and have test management or test automation knowledge, there shouldn't be a problem changing your role to something like SQA Engineer. You can always ask the company to change the role for you as well, but I'd simply put the role title that feels most correct when applying for new jobs.
As for non-resident employees, this will vary between target country and company. Many companies will explicitly state if they do not sponsor non-resident employees, but you may have to ask when applying.
As an aside, there are also different roles for QA roles focused more on test automation, and these tend to be called something like Software Development Engineer in Test (SDET) and tend to have a much higher focus on programming and test automation than traditional manual testing skills.