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Your are thinking correctly. In my travels I find the keeping personal data home and work data separate. Most modern servers record everything you do on their network & computers and use that against you.

Now with the iCloud page is the best way to see your personal email at work.
 
I wouldn't do it but that's me :)

They may have policies that frown upon such stuff.
 
I'm guessing it is because the work computer can be monitored. Correct me if I'm wrong.
There are many reasons why this could be bad.
  • It's against the policy of the place you work.
  • You have somehting in iCloud that doesn't belong on work computers.
  • Someone gets to your work computer (like IT) and mucks up what you have in iCloud (even if it's by accident)
 
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