I live in Ma, work in RI ,pay RI Temp Disability Ins. How do I report this on home state return - payments are by definition, a state income tax and eligible as a credit?

Yes, this is a credit on your MA return.  For TurboTax, this requires a double-entry.  

The first entry is on the W-2 screen in the Federal Taxes section of the program.  You need to make sure that Box 14 is identified in the TurboTax screen as a RI TDI payment.  The first screenshot will show you where this is entered.

The second entry is in the Massachusetts return.  If the w-2 screen is set up correctly, and you've prepared the RI return first, the RI TDI credit will appear with the taxes paid to another state credit.  Please see screenshot 2 below: 

It doesn’t add it for me even though I manually add it to my W2 (I take a picture, import, it doesn’t seem to like that field because it’s a mix of letters and numbers probably) 

I submitted mine without including it and it was a BIG difference in me owing MA vs them owing me. Thankfully I caught it before mailing it in!