**Response to Edited Question.**    
When you say 

> Compared to the other participants, the giver would have had to spend
> more, in terms of percentage, of in-game money he currently has.
> (think, beggar offering 5 coins vs rich man offering 5 coins kind of
> thing) For most of us who have been playing a while already, the
> budget is an affordable sum.

It leads me to believe that the giver most likely gave at the level he could afford.  

Now if he could not afford the minnimum then he should have either A) not joined the secret santa or B) messaged you as the organizer to ask for your advice.  

of course there is still a chance he simply misunderstood that it was a minimum.. 

I personally feel the best option in this case is to message the receiver and kindly let him know that the secret Santa has a minimum gift.  Something along the lines of 

> I just wanted to let you know that in our secret Santa we all get gifts that are at least $x.   I don't want you to worry,  I just wanted to make sure I communicated that well for next time.  

If you are concerned that the giver will take offense at that then you can generalize the message and send it to every one.  Which may be a good idea since it will show the receiver that you took action to make sure "un-fair" gift giving doesn't happen in future exchanges.  

If the give still doesn't meet the minimum next year (and after a personal reminder to give the minimum) then you may want to consider not letting him participate.  But that's a long way off yet.  

My original response to the original question is preserved bellow. 

**Original Response**
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.