Email Open Relay issue
This is a configuration issue with almost ALL email Postoffice servers.
Your Outlook (or Outlook Experes, or ANY POP3 Email Client) is configured for your
primary PrimaryEmailServer -- in your case Work or School (hum; this is an issue; see[1] below).
At home, you're attempting to send via the pathyourSystem-->yourISP-->yourPrimaryEmailServer
If this were to be allowed, then yourISP would have what is known as an Open Relay Server.
[1] see this article
Most reputable ISPs block port 25 going anywhere but their OWN postoffice server.
Solution?
1) Hopefully, your yourPrimaryEmailServer has a browser interface (also A.K.A. Webbased Email).
If so, you connect to it with any browser (at a special url), login with your email user/passwd and away you go!
2) You might be able to VPN into the Work or School systems and run Outlook as is