It complains that "This item could not be saved because Zotero is not open or is unreachable. Please open Zotero and try again." But I do not want to install Zotero to my desktop.
It's not currently possible to save a PDF directly to the web library (among other things because the web library doesn't support PDF metadata retrieval, so you wouldn't end up with a proper metadata item). For any significant work in Zotero, you should expect to use the desktop app.
I have installed the Zotero desktop app. But when I tried to sync in "Zotero Preferences - Sync - Settings", after tying in my User name and Password, then click on "Set Up Syncing", but then I got such error: SSL certificate error connecting to api.zotero.org
Upgrade Storage
https://www.zotero.org/support/kb/ssl_certificate_error