
Could be she has learned to be polite in her greetings and lying down is appeasing to the other dog. Could be submission, could be the sign of a quite confident dog offering reassurance to the approaching dog. Turid Rugaas calls the latter Calming Signals and if you google you might find a video or photo example on-line. I have her book.
The hair on her neck going up could be submission, aggression, excitement, apprehension, anticipation. From the little bit you've written she sounds quite well adjusted and I'd internet, unseen in person, bet on anticipation and excitement. :)
My dog does this and he is confident, sure of himself and out going. There is not an ounce of submission in him. I call it his "Alone and Unarmed" pose. Meaning he presents himself as unthreatening. I got one such meeting on video and you can see it here.
http://www.lab-retriever.net/board/labrador-retriever-videos/7411096-alone-unarmed-now-video.html