I agree with that. But also, I dont think he was only loyal becasue of lily. He did want the greater good.
If I was snape and i was "unloyal" then when Dumbledore deied i would have just stayed at hogwartd being headmaster/ Dark arts teacher. Cause voldy wouldnt doubt him and there still would have been a hogwarts. And, isnt that where snape always wanted to be anyway??
He didnt have to keep on with dumbledores orders. Now did he? HE may not even be dead if he didnt help albus.