I have almost the same problem.
My budgie (who I have for a year) loves coming give me kisses. It doesn't have outside cage time because I haven't made a budgie safe room though I plan on doing it one day. I say this to let you know that I don't whether it's a territorial issue or not.
When I talk to it, it jumps to the bars of the cage (from the inside) and kisses my nose and pecks on my glasses and eyebrows because I have my face close (I don't let her reach my lips). So she likes me.
But the hands? I don't know. If I get my hand near the cage while I'm doing this and try to put the hand close to the bars near my face while she's giving me kisses (slowly), she'll then focus on my hand instead and if she can reach the fingers she'll peck on them harder and harder until I take the hand away.
I've given her treats that I held from outside of the cage. I've done it also by holding the treat just at the entrance/door of the cage, in which only a finger will be trespassing the frontier between "outside" and "inside".
I don't have an answer but wanted to share my experience.
PS: I often try to put my hand near the cage so she can see it but not reach it, while talking to her. I don't have the regular "training time" but I kind of wonder if it would work...
She eats all the treats, broccoli, spinach, millet, but then she hops in my finger and bites.
I've seen the idea that she might be just be territorial. What do you guys think?
PS: Is it an occasion of stepping back? If I was doing the taming process from the beginning, and she shows comfort and stuff but the day I put my hand near the cage, she bites, does it mean I need to stop putting my hand inside the cage and go back to "hand outside" for more days? What if she always does it?