I think that this goes back to my point about what is expected on a local level.
A $2,000 deposit for a $800K home here would be a problem. If the buyer offer $2K at first and then came up upon request, it is likely that the listing agent and seller would be more skeptical about the deal going to closing.
This often presents problems from beginning to end. I don't want my clients putting more money at risk than need be, but I would rather them put a bit more money down and win negotiations on other fronts, which is easier with a stronger offer.