Let’s have a look at the translation of Summorum Pontificum 7 on the Vatican website.
A forward: Latin vult is from volo. Volo expresses volition, corresponding to German “wollen” and English “will”. In English it is rendered as “to wish, want, intend, purpose, propose, be willing, consent, mean, will”. With an infinitive it is “to wish” and, can equal something like in animo habere, “to intend, purpose, mean, design”… to have in mind. It can have the connotation of studere, “to try, endeavor, attempt” and even “to mean”, as it he doesn’t mean to do it”. It can mean, “to order, command”, or “consent, allow”. All the ripples of means deal with volition. It does not concern ability. It concerns intention.
Now look at the modern translations of SP 7 (emphases mine and comments):
Art. 7. Ubi aliquis coetus fidelium laicorum, de quo in art. 5 § 1 petita a parocho non obtinuerit, de re certiorem faciat Episcopum dioecesanum. Episcopus enixe rogatur ut eorum optatum exaudiat. Si ille ad huiusmodi celebrationem providere non vult [“does not want”, “wishes not”, “does not intend”, “does not order”, “does not choose” …] res ad Pontificiam Commissionem “Ecclesia Dei” referatur.
Italian:
Art. 7. Se un gruppo di fedeli laici fra quelli di cui all’art. 5 § 1 non abbia ottenuto soddisfazione alle sue richieste da parte del parroco, ne informi il Vescovo diocesano. Il Vescovo è vivamente pregato di esaudire il loro desiderio. Se egli non può [?!? “cannot”, “is not able”] provvedere per tale celebrazione, la cosa venga riferita alla Commissione Pontificia “Ecclesia Dei”.
Spanish:
Art.7. Si un grupo de fieles laicos, como los citados en el art. 5, § 1, no ha obtenido satisfacción a sus peticiones por parte del párroco, informe al obispo diocesano. Se invita vivamente al obispo a satisfacer su deseo. Si no puede [like the Italian – wrong] proveer a esta celebración, el asunto se remita a la Pontificia Comisión «Ecclesia Dei».
German:
Art. 7. Wo irgendeine Gruppe von Laien durch den Pfarrer nicht erhalten sollte, worum sie nach Art. 5 § 1 bittet, hat sie den Diözesanbischof davon in Kenntnis zu setzen. Der Bischof wird nachdrücklich ersucht, ihrem Wunsch zu entsprechen. Wenn er für eine Feier dieser Art nicht sorgen kann, [wrong… like Italian and Spanish] ist die Sache der Päpstlichen Kommission „Ecclesia Dei“ mitzuteilen.
English:
Art. 7. If a group of the lay faithful, as mentioned in Art. 5, §1, has not been granted its requests by the parish priest, it should inform the diocesan bishop. The bishop is earnestly requested to satisfy their desire. If he does not wish [like the Latin… “”if he doesn’t mean to”, “if he doesn’t intend to”, “if he doesn’t order…”] to provide for such celebration, the matter should be referred to the Pontifical Commission Ecclesia Dei.
This, friends, is why we check modern language versions against the Latin.
Summorum Pontificum is a juridical document. We need the Latin.
The next question is …
Why?
Why aren’t the languages consistent with the Latin? Whose choice was that?
Were I suspicious I would wonder if someone in the Secretariat of State wanted to give bishops of certain nations a way out, an excuse not to follow the juridical provisions of the Supreme Pontiff. “Eccellenza, we, a group of the faithful, have been requesting Holy Mass in the older form for years and don Abbondio won’t help us.” The Bishop, summons his minion and orders up a copy of the Motu Proprio. The minion brings it… in Italian, since His Excellency can’t be bothered with Latin. “See here, figlioli… the document says that if I can’t, then… well… I can’t. Non possumus, figioli. We can’t help you. Mi dispiace.”
Look. It is possible to over analyze the Latin or the other languages and go waaaaaay out into the weeds looking for complicated excuses for this or that variation. The simple answer is probably the correct answer.
UPDATE 12 August:
An alert reader sent this:
I located a USCCB newsletter from 2008 which, in a section entitled “Summorum Pontificum Formally Published With Minor [sic] Changes,” lists “Article 7’s phrase “providere non potest” (“cannot provide for”) was changed to “providere non vult” (“does not wish to provide for”)” as one of five changes made to SP between its initial release on July 7, 2007 and its formal publication on September 7, 2007.
http://nccbuscc.org/liturgy/innews/may-june2008.pdf
Here it is:

This could solve part of the mystery of the discrepancy. The other part remains: why are they divergent translation on the Holy See’s site?