Your 1-monads are called Parameterized monads by category-extras. Your 2-monads are a subset of Indexed monads (which have ALSO been called parameterized monads by some folks). Dan Piponi gave a nice writeup on them fairly recently, including a category-index writer monad that precisely matches your specification.
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chiming
Indexed and Parameterized Monads
Date: 2010-04-05 12:40 pm (UTC)From:-Edward Kmett