Are freshwater jellyfishes really jellyfishes?
The issue here is simply the meaning of the term "jellyfish" and the term "medusa". The first has no taxonomic value and, as such, can be used the way the book/zoologist/layman/journalist wants, unfortunately. Let's see it:

Cnidarians (Phylum Cnidaria) have two basic morphological types: medusa, or medusoid form, which is free-swimming, and polyp, or polypoid form, which is sessile:



There are Cnidarian species that display only polypoid forms and there are species that display both medusoid and polypoid forms in their life cycles.

Regarding the taxonomy of the most common Classes of Cnidarians, Hydrozoa displays both the polypoid and medusoid forms; Scyphozoa displays both the polypoid and medusoid forms; and finally Anthozoa displays only polypoid forms.

So, we can clearly see that Craspedacusta (an hydrozoan) is a medusa, or, in other words, that it displays the medusoid form in some point of its life cycle. There is no doubt regarding this:



The problem comes when we try to use the term jellyfish. 

Again, jellyfish has absolutely no taxonomic value. For instance, my teachers used to call an hydrozoan medusa as a "jellyfish". Some time later, when I started teaching zoology, I used to say to my students that an hydrozoan medusa is a jellyfish... that's not a big deal.

However, some zoologists apply the term jellyfish only to the scyphozoan medusae... in that case, jellyfish does have a taxonomic value. And, because of that, since Craspedacusta belongs to the Class Hydrozoa, its medusoid form cannot be called a jellyfish (again, if we accept the definition of those zoologists).

So, in conclusion, it depends on the book/zoologist: some say that that both Hydrozoan and Schyphozoan medusae can be called jellyfish, while some say that jellyfishes are only the medusoid forms from the Class Schyphozoa. That's the definition used by both the aquaticcommunity.com and the Nature Conservancy articles.

PS: Hydra is a Genus, not a Family. I edited your question just to realize that the mistake is from the original article, not yours.