TDIDT stands for "top-down induction of decision trees"; I haven't found evidence that it refers to a specific algorithm, rather just to the greedy top-down construction method. Therefore (seemingly) all the other algorithms you mention are implementations of TDIDT. The first iteration is due to Hunt, the "Concept Learning System" in 1966.
ID3 is due to Quinlan in 1979, improving upon the CLS. (Fun fact: it was originally designed to tackle the problem of deciding winnability of King-rook vs king-knight chess endgames.) This was further improved to C4.5, then to C5.0. This branch only works for classification.
CART ("classification and regression trees") was developed roughly in parallel with ID3, by Breiman, Friedman, Stone and Olshen in 1984. As the name suggests, this branch allows prediction of continuous variables.
The main differences between branches is how splits are determined, using different metrics. Early on it seems there were differences in handling missing data and such, but these appear to have mostly been included in all the more recent iterations.
Of course, there have been numerous other improvements or variations. Some (historical?) tree building algorithms generate nonbinary trees, some make splits on linear combinations of features instead of just one at a time, some try to reduce the greedy nature of the algorithm by looking ahead, some produce regression models in each leaf instead of constant functions, ...
W-Y Loh, "A Brief History of Classification and Regression Trees" (slides)
Lecture slides (Aida Nordman?)
Quinlan's 1986 paper
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