What kind of reductions are usually used in order to prove PP-completeness?
Here "PP-complete" means "complete for PP".  The definition of PP defines the complexity class PP; it does not define the type of reduction you can use, as that is an orthogonal concern.  So, "complete for PP" actually means "complete with respect to a particular class of reductions", and the meaning of that is implicitly parametrized by a class of reductions (usually the reader assumes you will be able to infer the class, from context).  For each class of reductions, you potentially get a different notion of completeness.

In this case, we're talking about polynomial-time many-one reductions, also known as Karp reductions -- the same class of reductions as is commonly used in the definition of NP-completeness.  This becomes immediately clear if you read the proof of completeness of MAJSAT (it's immediate to see that this is the type of reduction that gets constructed, in that proof).

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