I have several sites on how to line a bag, but it seems like websites aren’t helping you at all.
Is your main concern the sewing stitches showing at the top of the bag on the outside? I’ve only lined felted bags before, but I’ve sewn lables into non-felted things and I’ve been able to get the stitches to not show on the outside of the work.
I put the needle & thread through the middle of the acutal yarn, I don’t go around the whole piece of yarn. To do this I hand sew all linings into purses, I construct it with a machine, but sew it in by hand. Yes, this does take a long time, but I like the finished product better.
I’ve used clear nylon thread (that you can buy at JoAnn’s) for most of my linings, that way the stitches are even less visible. I do not use a “hem” stitch - where the stitches go in a straight line - (I don’t know what the proper term is) but I go into the lining of the purse, about 1/8" down, then go over the top of the lining into the stitches, so there is no “edge” of the lining for anything to get caught on.
I hope this helps, if not I can try explaining it some more and/or include some pictures if that would be helpful.