"Also, I'd rather choose the opposite path and keep the name "LanguageTool" for the normal version". I agree with Jan.
"About the page looking complicated: all I could think of is to move to explanation texts of the check mark/cancel icons to a popup." I disagree. If you put the explanations in popups, the information is not simpler. It is only more difficult to find.
For LanguageTool stand-alone, include a link to http://wiki.languagetool.org/software-that-supports-languagetool-as-a-plug-in-or-add-on.
The Comparison of LanguageTool Editions seems to conflict with http://wiki.languagetool.org/software-that-supports-languagetool-as-a-plug-in-or-add-on. In the comparison, LT is split into 3 groups:
* LT.org, LT for FF, LT for Chrome
* LT for LO/OO
* LT stand-alone
But, Software That Supports LanguageTool As A Plug-In Or Add-On is an list of all software that LT integrates with. It includes, for example, LT for Chrome, which on the comparison page is different to LT standalone. Change the organisation on the Software that Supports... page to be the same as on the Comparison of LT Editions.
"My interpretation is that a green check mark has a connotation of "good", and a red cancel icon means "bad"." My interpretation is that a green check mark means 'yes' and a red cross means 'no'. Possibly, remove the icons and use the words 'yes' and 'no'. Or, use the words and the icons.
For the line about additional homophone rules, the red cross for LT stand-alone and LT LO/OO is confusing because those versions of LT have the rules, but only after a manual installation of the rules. Possibly, use a third icon to mean 'yes, but'. (For example, I use this: ).