I see you don't have "require a password for system wide settings". It's better you leave that checked and enter a password whenever it's requested.
((off topic. I had the same issue (years ago) on debian linux with Firefox - but worse. If one simply left it open overnight - magically it's cache would get full and freeze the computer hard (so i kept closing ff, flushing cache, regularly, always closed when not using). later I compiled my "own nightly" of firefox and it never happened again (it was the binary, and being a binary i cannot say what was wrong or what was in it, who made it).))
First, you have to understand a web site designed for microsoft browsers might expect (or *demand*) that IE is being used. It's a competitive market and it's a shifting target, unfortunately. Because too many people are free to change "critical standards", you'll likely never see any computer in your lifetime that doesn't have issues with some websites that other browsers display fine.
You did a good job describing messages/logs you saw. And you already know (i'm going by what you said so far) that to avoid logout you only need to avoid one website. Note you have settings in safari to "act like another web browser" and or (show pages in an alternate manner). You might try some of those alt settings to see if that one website is now usable.
HOWEVER - to solve your real problem more is needed. what more could i ask?
please (all of you above!) specify exactly what web sites and pages are an issue so that others can "REPRODUCE" your problem. if the issue is not reproducible by others it may be very difficult to find help (excepting using xcode debugger, some really deep digging). also comment upon whether you think any "web plugins" are in use and what they are. if i can reproduce it, i may be able to see what is going on and even offer a fix.
give us a better picture of the logs (best is to cut and paste it, if you can). you mentioned some logs you saw. say what program you used to see the logs (did you say two different logs?), and show the log entries as exactly as you can (ie use screen shot, or jot down what you can and show it to us in a manner we'd recognize viewing the same logging program). were they all safari logs? were they two logs you saw in a list of all logs?
Your safari browser though is very good looks and very good multimedia (even 3d, sound) - and also it integrates apple's app features and hardware excellently. If you had a different browser (say, opera), you'd get none of that, or some but very slowly.