An example of the first type is the grandfather paradox, introduced by the french writer René Barjavel in Le voyageur imprudent (1943 - Future Times Three).
An example of the second type is The discover of Morniel Mathaway, a radio science fiction drama by William Tenn. It was originally transmitted by the show X Minus One by NBC:
A professor of art history from the future travels by time machine some centuries into the past in search of an artist whose works are celebrated in the professor's time. On meeting the artist in the flesh, the professor is surprised to find the artist’s current paintings talentlessly amateurish. The professor happens to have brought with him from the future a catalogue containing reproductions of the paintings later attributed to the artist, which the professor has come to see are far too accomplished to be the artist's work. When he shows this to the artist, the latter quickly grasps the situation, and, by means of a ruse, succeeds in using the time machine to travel into the future (taking the catalogue with him), where he realizes he will be welcomed as a celebrity, so stranding the professor in the "present". To avoid entanglements with authority the critic assumes the artist's identity and later achieves fame for producing what he believes are just copies of the paintings he recalls from the catalogue. This means that he, and not the artist, created the paintings in the catalogue. But he could not have done so without having seen the catalogue in the first place, and so we are faced with a causal loop.
(1) Deutsch D. & Lockwood M. (1994). The Quantum Physics of Time Travel, Scientific American, 270 (3) 68-74. DOI: http://dx.doi.org/10.1038/scientificamerican0394-68 (pdf)
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