Prosecutors will only amend a charge for two reasons: they are forced to because the facts no longer fit the requirements of the greater offense (1st degree burglary), OR as merely a negotiating tool in order to get a case off the desk and not gamble with losing at trial or simply to save the hassle of trial and move on to other things.
Each charge requires certain "elements." The list of elements required often differs by only one. For example, Burglary types often differ by the act being committed "at night" or "while persons are present."
The fact that the alleged victim is changing their story may or may not effect the prosecutor. The prosecutor can treaten the victim with charges of making a false police report if they now claim it didn't happen. However, if the victim is just not sure about identity of the theif, then that is different. Some prosecutors will try to negotiate a case to save the hassle of putting a wishy-washy witness on the stand.
Some times if a complainant retracts or changes their statement, the State may reconsider whether to prosecute at all, or whether to plea bargain for a lesser offense (depending on whether they believe they can make their case.)
Often plea bargaining includes not only discussing amounts of time in custody or on probation, but also charge bargaining - discussing the possibility of reducing the charge to a lesser offense. The decision to plea bargain is between you and your lawyer, and then it occurs between your lawyer and the prosecutor. The prosecutor is not required to plea bargain, but the always do in at least some manner to move cases. Some times a defendant will set a case for trial because s/he cannot get the bargain they want and their lawyer believes that setting the case for trial could result in a not guilty or a better bargain.