
Both my dogs will lie down sometimes when a strange dog comes into view. I always see it as them checking out the situation, kind of evaluating whether they are friendly or not. I remember them both being pinned to the spot once in a field by a Shepherd/Collie mix who was running rings around them. They were not sitting together, but the other (aggressive!) dog would not let them move. I can't tell you how mad I was when the owner eventually popped up and said "they don't want to play" !!!!!!!
BTW they are both Golden Retrievers, so not a small breed!
Oh, and they often have a "stand off" whereby they will both lie down at some distance from each other, then jump up and have a mad zoomie attack running around. They do this all the time with each other.
My male (the more playful of my two) will sometimes drop down, then get up and take a few paces before dropping down again, and repeat this a few times, when meeting a strange dog. He will then rush up to play (obviously if the other dog is making the right moves towards him).