You cannot use this machine for personal activities
That seems to be the policy, and from what you say it seems to be both clear and strict. You can't use the machine for your own activity. You seem to be confusing what counts as personal activity, however... you're talking about "Can I SSH onto my own machine and do work on my home machine?"
SSH'ing into your machine counts as a personal activity
It doesn't matter which machine is doing the processing, it doesn't matter where the files are being saved: the simple fact is that you are using this work laptop for your own personal activities/hobbies/etc. You've agreed not to do this (or possibly that anything done with their machine is theirs).
You are using the SSH software, without permission, for your own activity. The fact that your own activity is accessing your own machine changes nothing. Using an SSH application, or using whatever it is you want to use on your home machine, both are still things you are choosing to use for yourself.
You're still using their hardware, their battery cycles, their keyboard, their wear and tear. You're still doing your activities on their property which they have paid for and you have agreed not to do your activities on.
While I understand your point that these policies are typically put in place to prevent viruses, stop people using the machine during work hours for their own use, etc, that doesn't change the agreement or their right to tell you how you may use their property.
Quite simply, don't do it.