Correct, this is what everyone always hears about Burning Man. However, the idea that bartering happens at Burning Man is the most common misconnection. It is also how you can pick out that someone has never been to Burning Man (at least not recently).

I've been 5 years and from what I gather there was some bartering >10 years ago (not sure the exact date, but my impression is it was in the 90s). However, it shifted to a "gifting" economy where you give out gifts with no expectation of return and any Burner who hears that characterized as bartering would be quick to correct you.

Money isn't permitted except for the initial tickets, as well as coffee/ice (all of which are sold by the organizers).

It's otherwise a gift economy. Reciprocation is fine, except that the initial gift shouldn't be given with the expectation of reciprocation. It's a subtle difference, but it works out OK. Some would probably object to the term "economy" as well -- you're supposed to have arrived with everything you need to survive, so all the exchanging is just for fun.

Not quite - giving without any expectation of receiving is very different to bartering.