I figured out which of your sites it was based on the screenshots.
I cannot reproduce this here, or at your site using Internet Explorer 10 in normal use. However, it is possible that they have Compatibility View enabled for your site. That effectively puts the site into Internet Explorer 7 mode and the RTE doesn't work in that mode (deliberately). Aside from that, another explanation could be something else local to them. Script blockers, ad blockers being overzealous may cause the RTE to be disabled.
Ultimately, though, we can't reproduce this in normal operation.