Taxonomy: Categorising Collembola
The third column doesn't contain distinguishing characteristics at all -- they are the first word of the vernacular (or common) names for those groups on Bug Guide. Specifically, Bug Guide lists order Entomobryomorpha as Elongate-bodied Springtails, and family Entomobryidae as Slender Springtails -- so I think you're just looking at truncated data that's misleading you!

What you're looking for a taxonomic or identification key, which gives you set of yes-or-no questions that will allow you to identify an individual to a particular taxonomic group. Know Your Insect has a taxonomic key that allows you to identify insects to order Collembola, which provides exactly the sort of distinguishing characteristics you're looking for: "This illustration shows two characteristic features of members of this order: the collophore (center), a tube-like structure used to secrete fluids; and the furculum (lower left arrow)." Note that this resource refers to this group as an order, not a class -- that's not uncommon among different taxonomic resources. Wikipedia refers to this group as subclass Collembola!

Unfortunately, it doesn't look like there's a freely available taxonomic key for species within this group, unless you buy a specialized guide like A Key to the Collembola (Springtails) of Britain and Ireland.