My thoughts on each line Rogue:

“Come here, my little friend.”
If it cannot be done in a seamless way, I would just leave it. As my comments go this could end up being the “Obi being coy” reasoning.

“I haven’t gone by the name Obi-Wan since, oh, before you were born.”
I had a sudden thought on this. We know Obi Wan is “manipulating” Luke for his Destiny to train as a Jedi and eventually confront Vader, etc. If he did say something on the lines " … since you were born," I would think if I was Luke, that would be odd. Why would a person go by a different name since I was born? I would be curious and ask. So by Obi wan saying before he was born, that could help Obi wan avoid explaining this. That’s just my thought at the moment. Again, Obi being a liar again…

“I never seem to remember owning a droid”
Jedi’s did not have much possessions, basically their clothes and their lightsaber. So Obi wan was telling the truth, he did not own a droid. That line is fine for me.

“He [Owen] feared you might follow old Obi-Wan on some dan fool idealistic crusade, like your father did.”
Yep, just another straight up lie again. Again, part of manipulation. Luke does not know much about his father, and we know Owen probably gave vague answers or did not want to talk about it. So by adding these bits of lies about his father, it again manipulates Luke to following Obi Wan.

“Now, let’s see if we can figure out what you are, my little friend, and where you come from.”
All part of his coy-ness with Luke. The look Obi gave to R2 says it all that they are all “in the know” about keeping things from Luke.

I too favor the manipulating side for Obi Wan. Luke is the only hope, he has to have Luke involved, by whatever means necessary.

Your edit on the “Wanted you to have this” works perfectly well. Keep the manipulation going. Also, deep down, the Anakin he knew would have given him his/a lightsaber if he did not go to the Dark Side.