Look at the output from that second session. The first thing the server says is that it is
smtp.gmail.com. This means that your connection is being intercepted and redirected to your ISP's mail server, where, of course, your gmail credentials will not work.
Before you get that far, you should have had a certificate validation failure because the host name does not match the certificate name. If this was allowed to happen, you need to fix your config to always verify certificates as otherwise you are effectively the victim of an MITM attack and have given away your credentials.
In short, your ISP is blocking your outbound connection. While I can see you're using SwiftMailer, this is covered in PHPMailer's troubleshooting guide.
I edited out your ID and password from the SMTP transcripts; you should change your password now.