We are in fact being asked to send our EOS tokens away, out of our own possession.

This is not accurate at all. The act of registration is to inform the contract which EOS address your tokens will be available from on the native chain. The act of registering does not send your tokens anywhere, they remain in your wallet. It should be noted, there is no possible way to accidentally send your EOS ERC20 tokens anywhere by following these instructions.

Though I'm not sure what "claimed" would mean exactly, in this context.

Claiming tokens releases them from the contract to the address that you have contributed from. The claim transaction can only be called from the address you contributed from.

Have I in fact "registered" my tokens properly?

Without your ethereum address, there is no way to confirm this.

How am I to "confirm" that I have "registered" my tokens properly? How do I know this?

The instructions are above.

Can I still create this confirmation?

You don't create a confirmation, you read the state of the contract through the keys map exposed in the contract.

How do I re-gain verifiable control, custody and ownership of my own EOS tokens?

Based on what you have shared, I don't believe you have lost control, custody or ownership of your EOS tokens. I believe you are very confused, but with your public Ethereum address I would be able to tell you what's going on.

Why was it never made clear that "registering" my tokens is really MOVING my tokens to an address outside my control, that it's really a process of me sending my tokens to someone else?

You are very, very, very confused. Thousands of people have followed these instructions with success, and the act of registering does not forfeit your tokens in any way, shape or form. This can be verified objectively by reading the contract