Translation:I want you to let me pay for dinner.
I agree. There is no difference between "I want you to let me pay for dinner" and "I want you to let me buy dinner." There is no reason to favor cognates -- "pagare" and "pay" -- or to restrict the meaning of "comprare" to "buy." Either sentence in English is a good translation of the one in Italian.
In English, "to pay" does not have the same meaning as "to pay for."
Pay = give money to. One can pay a person: "I pay Lisa when she washes the car," which means "I give money to Lisa."
Pay for = buy. One pays FOR something one buys: "You pay for the cheese and I pay for the bread," which has the same meaning as "You buy the cheese and I buy the bread." So you would say "I want you to let me pay for the dinner," or "I want you to let me buy the dinner."
We can't say "I pay the dinner." It would suggest that I gave money to the dinner itself.
There is one funny exception: one can pay a fine, fee, or tax. "I pay a fee for parking," "I paid a fine for driving too fast," or "I pay the taxes."
I hope that is helpful!
grace, you don't need "that" & including it, makes the sentence sound very awkward. Also as mrlukens so clearly explains below, you shouldn't include "to". Where 'that' would be perfectly appropriate is if you substituted "wish" for "want" : I wish that you (would) let me pay for the dinner. Mrlukens' examples/explanations are very clear and correct.
lasciare leave, here" let" again Not permettere permit, ammettere allow accettare agree to consentire consent or allow. Just think let go is not as fitting a verb as these others.
Point of this verb lasciare being lasci are the -are endings. These stems are used in present, future, "would + verb" called present conditional and imperative. cercare cerchi and -ino dimentichi and -ino pagare paghi and - ino spiegare spieghi and -ino cominciare cominci and - ino mangiare mangi and - ino studiare studi and -ino Just trying to follow these conjugations and understand to recognize later.