Frequent use of word not found in dictionary, "programatically."

"Programatic" is a misspelling of "programmatic", which is in the dictionary. Your understanding of the technical usage is correct, and is slightly different than the common, dictionary definition.

I think the only reasonable alternative would be "automatically", since the programmer is automating the process, but this use is clearly inferior (at least to this programmer's ears) to "programmatically".

If we restrict ourselves to circumlocutions to avoid constructing useful and sensible words, then communication may well be impaired. In the case of "programmatically", I wouldn't even say that one has coined a new word. To anyone who understands the concept of using program code to achieve a particular result, the words "programmatic" and "programmatically" seem to me to be rather obvious constructions.

As a programmer, I have great respect for official documentation. In this case, however, I would say that the official documentation is incomplete, out of date, or has been misinterpreted.

The NOAD lists programmatic, and it reports it means of the nature of or according to a program, schedule, or method; one of the derivates reported by the dictionary is programmatically.

As alternative of programmatically, I can think of by (using a) script , by code, or by scripting.