I would like to hear the opinion of the Bitcoin community.

While I was researching different assisted wallets for The Bitcoin Hole, I came across a case that I'm not sure how to categorize. Here's the scenario:

An assisted wallet, where the owner and the service provider need both to sign a transaction to move the funds. You can think this a 2-of-2 multisig, where the service provider has one key and the owner the other. However you can also implement this with MPC (Multi-Party Computation). The implementation is not the issue.

I define something as "non-custodial" or "self-custodial" when these two conditions are met:

1- The provider can't move the funds without owner's permission.

2- The owner can access its funds without the provider's permission.

In the previous case, condition 1 is satisfied, but condition 2 is not. So, my conclusion is that this assisted wallet can't be categorized as "non-custodial" or "self-custodial".

Some questions I have in my mind:

- Is my conclusion correct?

- Is there any difference between "non-custodial" or "self-custodial"?

- Since the assisted wallet in this case cannot be considered "custodial" (as the provider cannot move the funds without the owner's permission), how should it be categorized?