Since this question has been unanswered for quite some time I will try to provide an acceptable approximation to an answer. Excuses beforehand for the many block quotes but it is not my area of expertise. The quotes are from a revealing article by Provine (1996).
As commented on by others, laughter can be specified better by their actual sound then by the written word (for a nice link on the bewildering variety of written laughter: http://forum.wordreference.com/showthread.php?t=419591). Spectral analysis of laughter in the lab showed that it is
[...] characterized by a series of short vowel like notes (syllables), each about 75 milliseconds long, that are repeated at regular intervals about 210 milliseconds apart. A specific vowel sound does not define laughter, but similar vowel sounds are typically used for the notes of a given laugh. For example, laughs have the structure of "ha-ha-ha" or "ho ho-ho," but not "ha-ho-ha-ho."
The Stereotypic structure of a laugh is, at least in part, a result of the limitations of our vocal apparatus.
The fake laughters may indeed be different, for example fake crying is also very challenging. Emotionless laughing is also characterized by different facial expressions.
As to where laughter comes from it is important to note
[...] that chimpanzees and other great apes perform a laugh-like vocalization when tickled or during play. [...] Breathy, panting laughter is probably the primal form that dates back to the common ancestor of all great apes and people. Human beings evolved their characteristic laughter after branching from an ancestor in common with chimpanzees (estimated to be around six million years ago, according to DNA hybridization data).
Laughter will, however, undoubtedly be affected by your surroundings, since
Laughter is a decidedly social signal, not an egocentric expression of emotion.
Hence, laughter is born out of an interplay of nature (evolution and restricions imposed by the vocal apparatus) and nurture (social setting).
Provine. Laughter. Am Sci 1996;84:38-45