Parameter vs. Member variables
Having a member variable implies that it will be holding state that needs to be held between method calls. If the value doesn't need to live between calls it has no reason to exist outside of the scope of a single call, and thus (if it exists at all) should be a variable within the method itself.
Style is always a hard one, once you develop one you can get stuck in a bit of a rut and it can be difficult to see why what you do may not be the best way.
In general, class members should represent state of the class object.
They are not temporary locations for method parameters (that's what method parameters are for).
I claim that it isn't a style issue but rather a readability/maintainability issue. One variable should have one use, and one use only. “Recycling” variables for different purposes just because they happen to require the same type doesn't make any sense.
From your description it sounds as if the other person's code you worked on does exactly this, since all other uses are basically covered by your list. Put simply, it uses private member variables to act as temporaries depending on situation. Am I right to assume this? If so, the code is horrible.
The smaller the lexical scope and lifetime of any given variable, the less possiblity of erroneous use and the better for resource disposal.