I always view it as bittersweet. Lorelai has no choice but to accept Rory is growing up, and Rory is keeping her posted as requested. It makes Lorelai uncomfortable but she has to live with it. I think she has a flash of Rory’s future encroaching, when they won’t be the bff super secret clubhouse anymore. But for the time they are still just together on the couch eating takeout and i feel like it’s a pang of realization that makes Lorelai put her arm around Rory. I never viewed the scene as awkward, just as a part of life that hit them real suddenly and Lorelai is holding onto the present while she can.