For Donation, that's the "tip jar" method. What you're saying is "I like what you've done, here's some money". This is usually seen as "general" monies for the project, and tend to go toward non-personnel operating costs (hosting, test hardware, etc.). There is no specific goal, other than the overall support of the project. See: Donate button on LMCE.org.
Sponsorships are when a (typically commercial) outside entity gets involved. They usually rely on the product and want to see it continued and improved. There may or may not be a specific agenda for a commercial sponsorship, but it's generally assumed that their involvement will steer the project in ways beneficial to the commercial entity, while simultaneously enriching the project as a whole. Hypothetically this would be represented by Dianemo paying a salary to l3mce for his work on LMCE. But I don't see this happening here at the moment.
I think there is some real potential for a Bounty program in LMCE. This can be as small as one individual and one developer agreeing to pay for implementation of a feature or piece of hardware, or as large as Kickstarter projects (which are a type of multi-donor bounty system). One good example is that I have a Vista 20p alarm system, but unfortunately did not get a VistaICM. I am no programmer, so I would find it invaluable to be able to pay someone to integrate either the new VistaICM replacement or the AD2USB device. Given some of the other posts in the forums, others may find such a driver useful as well, and if everyone contributes then everyone wins.
There are other funding methods, but they all have governance overhead that I think is out of proportion with the size of LMCE at this time. Take the "milestone" approach initially mentioned. As mentioned, you'd need to set up governance structures to ensure accountability and payment. And as the number of parties in each milestone transaction increase, you have more stakeholders, meaning more chance for disagreement and drama.
I like the idea for testing interest in forum posts (maybe a special sub-forum for this?), and if there is a lot, maybe creating a Kickstarter project. If only one bounty offer shows up, but is enough of an "angel investment" (able to realistically fund the entire feature themselves), then that could be a PayPal transaction between them and the dev. Maybe a half up front/half on completion payment arrangement on small transacts like that.