Easily confused with : all other Water-starworts. Their characteristics are very variable. The books say that they can most easily be differentiated from one another by the shapes and sizes of the seeds/fruits.
Autumnal Water-starwort (Callitriche hermaphrodtica) has never been recorded in Cartmell, nor has
Various-leaved Water-starwort (Callitriche platycarpa), nor has
Blunt-fruited Water-starwort (Callitriche obtusangula), nor
Intermediate Water-starwort (Callitriche brutia ssp. hamulata), nor
Autumnal Water-starwort (Callitriche hermaphroditica), nor with
Short-leaved Water-starwort (Callitriche truncata).
Pedunculate Water-starwort (Callitriche brutia) has been seen in the 4 tetrads which includes Cartmell but whether it has been seen in that same river is another unknown matter and, even if it has, then the leaves of that have in a 3-sided spanner-like notch at the termination, which these leaves shown above do not possess.
So, ipso facto, the above photos are indeed of Common Water-starwort.
Not to be semantically confused with :
Starwort Mouse-ear (Cerastium cerastoides) [a plant with similar name]
Identifying Callitriches is not easy because they have very variable characteristics, or so the books say. But Common Water-starwort is the commonest, and it is recorded as growing in this area. And it does look like it, so your Author is reasonably confident that this is Common Water-starwort, but if anyone thinks otherwise, please drop me a line and tell me what you think it is instead.