What does "sponsor" mean here?

Question

The following is a passage from an article on the idiom kill two birds with one stone:

Some people feel that this term is a bit negative, given the association with hunting and death, and they prefer more positive twists on the saying. Several organizations have even sponsored contests to come up with a new and more animal-friendly version of “to kill two birds with one stone.”

What does sponsor mean here? Is it sense #1 from OALD:

(of a company, etc.) to pay the costs of a particular event, programme, etc. as a way of advertising Sports events sponsored by the tobacco industry.

or sense #2:

to arrange for something official to take place The US is sponsoring negotiations between the two sides.

Answer 1

Sponsor could simply mean "propose and organize something" or "doing something to make something possible."

For example, in your case, it could be that those organizations made available the space where the contexts took place (e.g. one of the buildings owned by a organization).
When a company is sponsoring a public domain software project, the company could make available one or more of its developers that would dedicate some of their working time to develop the code necessary for the software project, and provide the patches necessary to fix bugs.

Answer 2

Probably #2, though possibly also #1. When we say that someone "sponsored a context" we usually mean that they planned and organized it. Sometimes they pay for any prizes themselves; other times they get someone else to pay for the prizes. Like when a charity organizes a contest, often they get local businesses to provide the prizes. In that case we could say that the charity sponsored the contest in the sense that they organized it, and also that the business sponsored the contest in the sense that they paid for the prizes.

You'd need more context or elaboration to say exactly which.

But in context, I think they're talking more about organizing than paying-for here.