I would not say anything to the giver. He gave what he thought was appropriate according to the rules/social expectations that he was aware of.
I would apologize to the receiver, telling them something along the lines of "I made a mistake in planning and have learned a valuable lesson, that next time I will be sure to set a maximum and minimum amount."
If the receiver continues to complain just repeat "I'm sorry I'll make sure it doesn't happen in the future."
Good luck
PS
While I answered your main question, I failed to answer some sub questions so here goes:
do I tell him that the recipient got upset?
No he gave correctly according to his understanding of the rules. There is no point in making him feel bad for something he didn't know about.
How do I tell him to send something more appropriate (up to budget), without upsetting him, too?
Don't tell him directly next year just be more specific with the gift instructions stating. That the gift should be between $x and $y amount.
Edit #2 It is my understanding from the question that the receiver felt that the budget number was a minimum (you must give at least this much) the giver saw the same number as a maximum (you must give no more than this amount) So they both acted correctly with in their understanding.
Further more gifts do not have to be, and in fact they are rarely ever equal in value. The point of gift giving is not for me to give you $20 of something and you to give me $20 of something. But rather for me to thoughtfully select/make something I think you will appreciate and give it to you hoping that it brings you some joy. It just so happens that human beings have evolved to have a strong sense of reciprocity so we frequently try to give gifts of equal value. But that reciprocity is not perfect and can vary from situation to situation.