I'd settled on an EM-10 III. Why do you think the II is a better buy?
Olympus changed the menu system between the II and III. They moved a couple of useful features like bracketing and HDR into the Auto modes out of P/A/S/M IIRC.
They also removed some flash control stuff but I never did any off camera flash so have little idea if that matters.
The III got 4K video
The III was basically a firmware update.
The IIIS adds sweep panorama if you're interested.
The IV got the more modern 20MP sensor and a flip down screen which I would like if it weren't so expensive in comparison to my II.
Which E-M10 should you buy? We pit the Olympus OM-D E-M10 Mark IV vs Mark III vs Mark II in this comparison
www.digitalcameraworld.com
I bought my II in 2018 used and haven't seen anything better in the small, cheap, light, powerful market.
The E-M10 series has a built in pop-up flash - the M5 would be worth considering for me if it had that too.
Its really not much larger or heavier but it is weather sealed with the right lenses.
Worth pointing out that the 12-40mm Pro on the M10 casts a shadow so it's use for fill in is less helpful than using a smaller prime or zoom.
Where I am looking for some help is on lenses. I've set an initial outlay of £250-£300 on glass (new or used) and then add to it month on month.
If you want primes I'd strongly recommend the Olympus 17mm f1.8 as a walkabout lens.
If you can find one then I'd go for the Olympus 12-40mm F2.8 PRO as a single, general purpose lens.
Since I got mine the primes make very few appearances.
The Panasonic f1.7 25mm is a bargain new or used IF you like that 50mm equiv. focal length.