It is first worth noting that Karol Wojtyla (the future John Paul II) explicitly cites this as a medieval distinction. You are absolutely right that desiderium can mean desire, but he is not just giving a Latin translation of a common term: he is appealing to a traditional distinction that he hopes will shed light on the different ways love (amor) can be understood.
I cannot find this distinction explicitly made in the works of Thomas Aquinas, but his contemporary Bonaventure (1221-1274) appeals to it in his commentary on Lombard's Sentences, book I, distinction XVII, part I, article I, question III. (Yes, Sentences commentaries were labyrinthine!)
Here's the relevant part from the conclusion:
Ad intelligentiam autem obiectorum notandum, quod triplex est amor, scilicet amicitiae, quo aliquis optat alicui bonum; concupiscentiae, quo aliquis sibi desiderat aliquid; et complacentiae, quo aliquis requiescit et delectatur in re desiderata, cum praesens est.
Dico ergo, quod caritas amore amicitiae nullo modo est amabilis, quia non est beatificabilis; amore concupiscentiae est amabilis ex caritate, secundum
quod diligens desiderat amplius diligere; amore vero complacentiae, scilicet quod diligens acceptat ipsam dilectionem, qua Deum diligit, hoc diligenda est caritas, secundum quod iam habetur, eo quod ipsa est bonum valde acceptabile.
My quick translation, with technical terms kept as English cognates:
In order to understand the objections, I should note that love [amor] is threefold, i.e. [love] of friendship, by which one wishes good for another; [love] of concupiscence, by which one desires something for oneself; and [love] of complacency, by which one rests and is delighted in the desired thing, when it is present.
Therefore, I say that charity [caritas] is in no way loveable by love of friendship, because it cannot be made happy; it is loveable with the love of concupiscence out of charity, insofar as one who loves [diligens] desires to love [diligere] more; but charity is to be loved [diligenda] with the love [amore] of complacency, i.e. that which, loving, accepts that love by which God loves, insofar as it is already possessed, since it is a very desirable [acceptabile] good.
(I am not sure why Bonaventure uses the neuter relative pronoun for amor, but it's clear that that's what is going on.)
How well that maps onto Wojtyla's way of setting up the distinction is a different question. My initial impression is that desire is an intuitive way of understanding amor concupiscentiae (given that the noun itself is related to cupio); I'm less sure about translating complacentia as "attraction," but he means it in a pretty specific way.
Regarding the grammatical point about the genitive, this is a standard construction of the so-called "appositional genitive" (see Allen & Greenough § 343.d). English uses the same construction, e.g. "the city of Rome." Strangely, the edition I see of Love & Responsibility has amor concupiscentia (two nominatives), which seems incorrect to me. Given that you cite it with the genitive, perhaps it was emended.
