Exploring Electrum and see that I can add a description when generating a receiving address.
The description cannot be edited later... would you happen to know why? And is that description public domain?
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I don't know why the description cannot be edited, that sounds like an implementation quirk. You could try looking at the Electrum documentation or inquiring on the Electrum GitHub project, if nobody else here can tell you.
The address description is your reference to identify what invoice the address belongs to. Some wallets include the description when sharing the address as QR-codes, or even have separate fields for your reference and a message to the sender. Generally, wallet addresses are neither registered nor gossiped on the Bitcoin network, they are just transferred out of band from receivers to senders when the receivers indicate how they'd like to get paid.