A deed in lieu of foreclosure is where you sign the house over to the lender in exchange for them not formally filing forclosure. The benefit is that generally the deed in lieu satisfies the loan, whereas a foreclosure might result in a deficiency sale price and you still owing money on a home you no longer own (some states prevent this).
If you filed BK but continued to make payments, and the mortgage company accepted the payments, then they must not report that they payments or late or not made. However, if you did not make the payments, and you did not reaffirm, did you include it in the BK?
A short sale is a hybrid between deed in lieu and foreclosure. You sell the home for less than what the loan amount due is, and the lender accepts the sale as full satisfaction for the loan. You must get the lender to agree to a short.
Coming out of BK, with a current mortgage showing missed payments will prevent you from getting another mortgage. I suggest renting until you get things worked out, unless you can salvage your current home situation.
