You needn't give your precise title in your resume. People reading it will not pay much attention to exactly how you title yourself, unless you bill yourself ridiculously, i.e. Amazing Genius Senior Engineer!. (In which case you may well be eliminated immediately.) They will look at your background and experience in considering you for hire, not at how you title yourself.
If you are indeed a senior engineer , bill yourself as such in your resume. Don't say (senior equivalent) or (flat org) - sounds apologetic or desperate, if not dishonest.
Be concerned about presenting your background, your skills and your experience accurately in your resume. Those are the important things - not the title - which can be rather meaningless, as your own experience shows.