Well, that sounds like the files are being downloaded fine. For the completer piece of mind, just check the maps downloaded at your current zoom level (MyDocs/.maps/OpenStretMapI/18/...) are shown correctly in Photos. Otherwise it looks like a render problem in Modrana.

The last thing I would try before blaming Modrana is, delete all the maps (just rename OpenStreetMapI and OpenStreetMapII to something else while Modrana is not running), then start Modrana and let the maps download again. There may be some weird caching problem.

I don't see anything suspicious in your log files, BTW, but I am hardly an expert.