In LibreOffice version 18.104.22.168 I installed LanguageTool version 5.1.3. This seems the last user version. The newer versions seems to be developer version. I tried that before, it failed, and now I use 5.1.3. My OS is Ubuntu 20.04.
The installation seems to be ok. In LibreOffice Menu -> Tools -> Extension manager -> I see LanguageTool-5.1.3.oxt without an error message.
But I do not see a window with 4 LanguageTool icons to check my text. To verify I copied the text, with spelling errors, in the Dutch language, from https://languagetool.org/ in LibreOffice. But in LibreOffice is do not see all spelling errors as I see on your website. So it seems the installation is not correct. Is there a way to check this? And, if so, hopefully correct the installation.