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I think you want to know what ssh-add does, and why the first way to use ssh-add failed while the second way succeeded. I'm answering with that assumption in mind.

Why ssh-add works as it does

ssh-add is used to load specific identities in your ssh agent. You have several identities available to you, and your ssh agent will offer them to any server you try to connect to, in an order I don't fully understand.

Under typical circumstances, a given server would only accept one of your identities. For example, let's say I added id_rsa_server1 to connect to host1, and id_rsa_server2 to connect to host2. When I connect to host1, ssh offers id_rsa_server1 to the server and connects successfully. When I connect to host2, ssh first offers id_rsa_server1, which fails, then tries the next one, is_rsa_server2, which now succeeds.

Your problem when you loaded the two identities is that github.com knows both helen and snow. You want to log in with snow, since that's the account that has permissions on that particular repo, but the ssh layer succeeds with the first accepted key, helen, which doesn't have the required permissions.

So when you unload your identities and load only the right one, things work because this times you log in to github.com as snow, with the required permissions to do the push.

I hope this helps clarify what is going on.

Alternative and recommended solution with .ssh/config

Because you're connecting to github.com by that name, I think your .ssh/config does not do anything. It would only do something if you connected to github.com-Snowbabe as the server name.

This should work:

.ssh/config with User=git: