The hot topic of the moment is how to brand the neutral ATProto account in a way that it understandable for non-technical users. Here is my quick idea:

"Log in with your .social account"

or maybe even with your ".soci@l account", if we want to keep the reference to the AT. The @ however, is quite noughties marketing cringe (maybe we can have that idea in the logo, but I am mainly good with words, and rather bad at graphic design).

Log in with your .social account

Your .social account is your personal account for all the social things you do on the internet. You might already have one when you signed up for Bluesky and got something like yourname.bsky.social. Maybe you have one from northsky.social or eurosky.social or blacksky.app.

You can use this same account to sign in to many different social apps. Your identity is not locked into one app or company. Many different companies and organisations can offer you such an identity and they work all together to keep you identifiable as the same person. In the background they give you a unique number, called did, that looks something like did:plc:44ybard66vv44zksje25o7dz. It is similar to a social security number, that points uniquely to you, but for the whole internet. It is a practical identifier for computers, but difficult to remember for humans. That is why the social internet also uses readable aliases in the form of web domains and subdomains. When you signed op for Bluesky, you got such a subdomain for free, something like yourname.bsky.social. You can always change it. If you are a bit tech savvy you can even change it to a domain you own, like mycooldomain.com. However, the domain will always refer to the same did (your online 'social security number').

We call it a .social account, because you use it for everything social, much like you use a mail account for sending and receiving mails. Many providers use the .social domain ending (like bsky.social, northsky.social or eurosky.social), but that is not a must (e.g. blacksky.app or tngl.sh are as valid as 'social accounts'). It can be any web domain, but it will always be social.