Moral to the story.... do your best to document the scene before leaving. Take pictures/video, prepare a drawing, get names of witnesses (if any). The physical evidence is not going to change, unlike a persons recollection (either intentionally, influence of others or passage of time).
Police reports are not as helpful as you might think. Often times it will be difficult or slow to get police to respond to a non-injury accident. And, all they do when they get there is gather information. Officers may prepare a report and MAY suggest who was at fault. If there is no conduct that violates a Vehicle Code, there often times will be no conclusion of fault.
Your thinking should be: what information would be useful for a judge, arbitrator, insurance representative, etc., in determining who caused the accident?
DO NOT record conversations unless you have first advised the person that you are doing so.
Don't assume that someone's "promise" or "admission" of fault is going to be presented unless documented in a signed writing. Your discussions afterwards are not likely to carry much weight and, as you experienced, the story changes.
The best strategy is to document your conversations the best you can. If the other side is not inclined to chat by e-mail or letter, you can prepare a letter after an oral discussion that claims to "confirm" the discussion. Close the letter by stating that if the above does not accurately reflect the conversation, that you should be contacted immediately.