I'm on an iMac + High Sierra (Version 13.0.3 (14608.3.10.10.1)) and the latest Zotero (5.0.81-beta.2+bbbd02444).
but when you open a webpage with references nothing happens
I was able to reproduce the extension not showing up in High Sierra. We'll investigate.
I had to deal with household stuff (Son, chickens, lunch) and when I got back to the computer I restarted both Zotero and Safari and this time the icon appeared but didn't work, but the extension had re-appeared, so I unchecked it, checked it again, and now it is working again, Phew! Have successfully downloaded another paper to Zotero!
Have you tried restarting your computer?