No, that's not really related. Cross-site scripting's name comes from the vulnerabilities which allow an attacker to insert a <script> tag pointing at a script

No, that's not really related. Cross-site scripting's name comes from the vulnerabilities which allow an attacker to insert a <script> tag pointing at a script on another domain (or an inline script). It doesn't have to do with cookies and doesn't get around or really interact with the "block 3rd party cookies" setting.