"ProConnect will usually generate the other state tax credit for state returns whenever you enter taxes withheld or paid to another state anywhere on the return. The taxes are usually entered on the income screen they're related to (like a W-2 or a K-1) with the appropriate state identifier selected, like these fields for a partnership K-1:"

I hope that is not how it works since that would be wrong.

Just because I have withholding on my W-2 for a nonresident state does not mean I paid tax to that state. I could get it all back by filing a nonresident return. Or I could get some of it back.

"The 309 in particular looks like it will only generate when there's both income taxed to another state and taxes paid"

This is correct.