As you know my choice was a Sony. The A6000, is still well thought of, but for me out of those two choices the 16-50mm & 55-210mm lenses will be more versatile than those bundled with the Olympus. That would be my choice of the two, but I reckon either would be a good choice. The Olympus does have image stabilisation - not sure if it does much the same as on mine, but if it does it is well worth having. I can shoot 1/40th second hand held with sharp results.
It may be worth enquiring about the cost of the A6000 body + the 16-
70mm lens. .... but as it's a Sony Zeiss lens it may be above your budget, but if you can afford it it is a very good lens. I have 5 lenses, but use the 16-70 90%+ of the time. It has lens image stabilisation as well as the in camera stabilisation. (God knows what that really means, but it seems to work!) I do have the 55-210 lens - it's actually one of the cheaper Sony lenses, but nevertheless is good quality & one & well worth having. I have added a x1.7 telephoto extender to it (an Olympus one) - a tad cumbersome but gives up to almost 360mm zoom, retains surprisingly good quality & for a fraction of the cost of buying a longer Sony Zoom.
Just on a side issue, whist in Tassie I trusted an old friend, an ex photo journalist, with my camera - just to let him have a look at it. Somehow he managed to get a droplet of what I suspect was beer onto the sensor after he removed the lens. I didn't discover it until several days later. Result was an ugly back spot on all my photos which took a lot of post editing to remove. Annoying & tediously time consuming.
I almost tried cleaning it myself having bought a sensor cleaning kit, but, sensibly I believe, chickened out & waited until we were home to get it professionally cleaned. I was worried that after almost 3 months on the sensor it may have been difficult to remove without expensive damage, but took it to specialists at 'Camera Clinic' in Easey St, Collingwood. Drove down there a couple of days ago & they did a fantastic job on it in an hour & cost was a very reasonable (I thought) $77. Can't recommend them highly enough.
http://www.cameraclinic.com.au/index.../imager-cleans