Regarding botanical naming, normally, I would quote The Plant List; but I think it is unhelpful as regards the variations of Fagus sylvatica. Instead I will go with the nomenclature and spelling used by the RHS - Fagus sylvatica var. heterophylla 'Aspleniifolia'
Cut-leaved European beech is a variation that arises naturally now and then, and as such, there are undoubtedly different clones in existence. Some of those might be slighly different from others, eg. in the degree of incision of the leaves. Some of those might be worthy of cultivar or varietal names; many won't be. Partly because of such history, the tree goes by several botanical names - various permutations of heterophylla, 'Aspleniifolia', 'Asplenifolia', laciniata, Heterophylla Group. Some of those are correct, some wrong, some synonyms, some slightly generalized. It is all very confusing, and because of that, references themselves may be inaccurate, compounding the confusion.
This is an example of a common name working better, for general purposes, than its botanical equivalent(s).