Invalid security certificate for my website host-they say the problem is Apple Safari and use Firefox instead

For the past few days, I keep getting an invalid security certificate in Safari whenever I select Edit My Site from my website homepage (http://annaporterartist.com), or whenever I select anything requiring a secure log in from my website host main page (FASO.com). I have contacted technical support at my website host (fineartstudioonline.com) and they say that this has been an intermittently recurring problem in Safari for years and they recommend that I use Firefox instead. As proof of this they emailed a link to an Apple Support discussion, but it was for Mac OS X Lion v 10.7.4 and Safari 5.1, even though I told them I am using Mac OS X Mountain Lion v 10.8.2 and Safari 6.0.2. I do not get this error message anywhere else on the web using Safari. I did try Firefox and it seems to work fine, but I prefer Safari and I want to know why Safari is not working as it should be. I am concerned that there is a real security problem with my website host and I need someone to explain why I am getting this error message, what it means, and if it is, in fact, a known problem with Safari or is my website host corrupted? Really tired of technical support playing pass the buck or pretending the problem does not exist.

The specific error message is:

Their response to my inquiry and my reply is shown below: