Thank you all for your very helpful replies.
I now feel more confident with Bessie's behaviour after hearing from you all that this is not unusual. I have had several dogs during my life none of which have exhibited this behaviour, which is why I asked for advice.
On our "short" walk in the local park we meet a lot of dogs who play in a group, so I shall now let her join in without worrying.
Regarding her breed, I said before she is just "Unique". Maybe some Lab, but she hates water and is not greedy at all. I think she just has the best traits of all the breeds that have contributed to her DNA.
She is a wonderful girl and we love her.