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|TI-83 Basic||68k TI-Basic||TI-Nspire Basic|
TI-83 Basic is the most commonly used, because the TI-83/84 series has been heavily marketed by Texas Instruments to high school students needing a graphing calculator for math and science classes.
At the same time, it is the least powerful language, lacking many of the complex commands and programming capabilities.
68k TI-Basic is much more powerful than TI-83 Basic, with support for calculus, indirection, local variables and functions, advanced picture manipulation, and several other features that make it a very rich language.
However, it isn't used nearly as much as TI-83 Basic, so it doesn't have as big of a community developed around it.
TI-Nspire Basic is quite similar to 68k TI-Basic, but not as programmer friendly: it has poor I/O support, rigid constraints on program execution, and documents are used instead of programs.
In addition, it is still relatively unknown to the TI community, so there isn't much documentation available yet.