So for all intents and purposes, there is no difference in terms of stabilization between all recent Olympus OM-D cameras (even the budget E-M10 series has 5-axis IBIS now since Mark II).
I believe this is not true. EM5-II is no longer a new camera (two years old). I have both an EM1-II and EM5-II. I did have an EM5 and believe me, its IS was nowhere near the EM1-II. It is really a night and day difference in dark places like interiors or cities or towns after dark if you don't want to use a tripod.
My notes:
"The EM1 II has 6.5-stop stabilization. The in-body stabilization itself gives 5.5 stops, and the Sync IS gives 6.5 stops with OIS lenses. E-M5II offers 5 stops and E-M1II offers 5.5 stops, EM5 has 4 to 4.5 stops."
Notice the apparently "small" difference between the EM5 and the EM1 II. "only" 1 to 1.5 stops. My personal experience is this is all the difference in the world if you want to do low light photography without a tripod. Not only am I able to take evening and night photos of harbors in Rockport Texas hand held whereas I could not do it with my EM5 or my Sony A77. I'm talking no special lens because I just got my 120-100mm a couple of weeks ago. In New Orleans last week I was shooting with the EM1-II and the 12-100mm lens all the time, including of course interior shots and evening and night shots. Similarly, I was unable to take good night shots handheld with the EM5 or the Sony A77, even though the A77 can combine multiple shots in an attempt to get a better result.
I'm attaching an evening photo that I was unable to take with my EM5 or my Sony A77 (both have in-body stabilization-- something I demand in my cameras). Even with serious post processing I could not get a good enough photo to sell.
I thought long and hard about the extra money for the EM1 II, and then for the rather expensive 12-100mm lens. I'm very happy with my choices. Of course you may make other choices, but for someone who hates tripods and likes iffy lighting, this is definitely a game changer. Playing around, I have hand-held 2 seconds with pretty good results with the EM1-II with an ordinary lens, even without attempting special stability like leaning against a post or something. It is so good that I have to remember to not let it take the ISO too high because that is more detrimental than a slower shutter speed. I should add that apparently I'm pretty stable and some people cannot hold a camera stable for one reason or another... Since I'm now 76, I'm not sure how long my stability will last... I guess I can start using a monopod/cane when I get shakey...

Back of St. Louis Cathedral, New Orleans. 1/5 sec, ISO 800.
--
Judy