Hi Alan,
You asked "Out of all the sites I have visited, and like most people I suspect that is hundreds, why is it only the ME site that is a problem?"
Well, I agree with Clivel who said earlier "So it would seem that the problem is not caused by your choice of OS and browser, but is instead caused by something else on your system, Until we can establish what that is, it is extremely unlikely that Katy will be able to duplicate the problem so as to create a fix."
I think you have a combination problem in the sense that "your system" includes everything between you and the ME Site. I suspect that the ME Site is sending instructions to your browser that are touched by something else such that your browser misplaces the black bar. It is likely that when other sites malfunction you don't notice.
The services behind apparently simple web browsing are complicated and fluid. For example the ME Site's login page calls 9 different web sites all of which are capable of asking your browser to do things that might not work properly. ( The sites are http://www.model-engineer.co.uk, http://www.gstatic.com, http://www.google.com, http://www.google_analytics.com, http://www.facebook.com, static.ak.facebook.com, s-static.ak.facebook.com, connect.facebook.net, and ads.myhobbystore.co.uk )
Software and human errors are common on the various systems that comprise the Internet. As a result your browser may be asked to do stupid things and the way these are handled (or not!) might well cause the kind of problem you've got.
You might like to try right clicking the offending page and selecting "Inspect Element". This opens up the detail that Firefox is working with. You might see an error that relates to the black bar, though decoding this stuff isn't for the faint hearted.
My example doesn't explain what you're getting but Firefox reported the following errors and anomalies on my view of the ME login page:
Net Errors
GET http://ads.myhobbystore.co.uk/adserve/;ID=154425;size=728×90;setID=162072;type=iframe [HTTP/1.1 200 OK 338ms]
carouFredSel: No element found for "#CarouselMain". jquery.carouFredSel-6.0.4.js:4261:4
GET http://www.model-engineer.co.uk/wp-content/uploads/sites/4/images/website_images/Header980x222.jpg [HTTP/1.1 404 Not Found 34ms]
Password fields present on an insecure (http://) page. This is a security risk that allows user login credentials to be stolen.[Learn More] <unknown>
Password fields present in a form with an insecure (http://) form action. This is a security risk that allows user login credentials to be stolen.[Learn More]
Console
Error in parsing value for 'background'. Declaration dropped. default.css:2387:13
Error in parsing value for 'background-image'. Declaration dropped. default.css:2389:22
Expected colour but found 'top'. Error in parsing value for 'background-image'. Declaration dropped. default.css:2390:38
Expected ',' or '{' but found ';'. Ruleset ignored due to bad selector. default.css:2493:69
Expected 'none', URL, or filter function but found 'alpha('. Error in parsing value for 'filter'. Declaration dropped. cropper.css:23:8
Expected 'none', URL, or filter function but found 'alpha('. Error in parsing value for 'filter'. Declaration dropped. cropper.css:48:8
Expected 'none', URL, or filter function but found 'alpha('. Error in parsing value for 'filter'. Declaration dropped. cropper.css:97:8
Expected declaration but found '*'. Skipped to next declaration. jquery.bxslider.css:21:1
Unknown property '-moz-box-shadow'. Declaration dropped.
I see that you've eliminated the business account as a possible cause. Can you think of anything else in your configuration that's has been in place for a long time? It's a very odd bug in that "This problem is not new. It was in evidence when I experimented with Debian, OpenSuse, Centos, and Ubuntu all of which was several years ago." Also "One of the three machines (not working at the moment) has had a fresh, default install of LM and the first site I visited was the ME one." As your configuration is respectable, whatever is causing this bug might be very difficult to pin down. I've run out of ideas, at least for the moment.
Please try a different Browser – I think the site is worth it.
For those who prefer news of hands-on engineering I confess wasting yesterday afternoon making a complete horlicks of a simple brass screw. Mistakes included: misaligning and over-tightening a knurling tool so that it twisted off the tool-post; carefully turning the rod down to 2.75mm ready for an M3 thread, mangling the finished knurling on an otherwise perfect second attempt during parting off; and then breaking a digital caliper by dropping it. The root cause throughout the fiasco was operator error. I gave up in disgust and a barrage of bad language.
Cheers,
Dave