To answer the question you actually asked, if the charity has mentioned your name to the school without you having agreed that they could, then they absolutely have done the wrong thing. If you had agreed with them that a team within the charity could come back to you with some ideas, that is what should have happened. Whether the message to the team got lost in translation or whether they are very amateur in their operations, this is not OK.
I can see that whether your plan was good or bad, over the top or not, you’re now in an embarrassing situation and you’ve not had the chance to consider the information provided by the charity and how you might raise it (or not even raise it at all) with school.
I think you should write a short and to the point email to the most senior person at the charity explain what has happened, make clear that you did not ask them to contact the school and you never gave them permission to pass on your name. Make clear that you are writing to raise an issue about their internal processes, and that on no account are they to contact the school again or discuss you or the enquiry you made. If they are any good, they will use this information to make sure that this doesn’t happen again. But don’t pursue it and turn it into a new crusade!
As for what you do with the school, I think I’d say that the charity got in a muddle, that you weren’t criticising the school, you are just very interested in the charity’s work and you wanted to see what they had on offer because you were thinking of mentioning the charity to the teachers if they seemed to have good ideas/resources. And whether you should have done all this in the first place or not, just back off and give up on the idea.