I'm not sure this is really a distinction with a difference...so you can be an illegal, roll into a voter registration agency and register, but only for federal elections?

No. If you're not a legal citizen, you're not allow to register for or vote in federal elections.

Arizona back in the day tried to add a requirement to the process of registering that you had to provide documentation proving your citizenship at the time of registration in order to register. The Courts struck that down. There exists a federal statute that outlines what documentation you have to provide in order to register to vote, and the Courts held that Arizona couldn't add to required documents. It was pre-empted by federal law.

So Arizona tried to work around that, by adopting dual classifications of registration. They argued that the federal statute only applied to federal elections, and that AZ was free to adopt more stringent documentary requirements for state-only elections. They have also adopted legislation that tried to use some legal legerdemain to get around the federal registration statute and prevent people who didn't independently get their citizenship checked from voting by mail or voting in the Presidential elections.

The dual state-only election citizenship test was permitted by the Court to go into effect pending the litigation. The rest was not.