You are correct, Obama signed the budget and pushed his $787 million stimulus that nearly destroyed the country by increasing unemployment. You can't redirect that kind of wealth without major dislocations as Bush had already demonstrated with the TARP and the bailouts. Obama's stimulus, like Bush's TARP didn't get spent until the next year.
If the TARP didn't hit the 2009 budget then just where would you put it? The actual money spent was authorized by Bush. 2009 was his disaster. 2010 was Obama's disaster.