What is a general term for a program similar to Texter, AutoHotKey, PhraseExpress or the shortcut functionality on the old Palm OS?
Input method is close, though it's usually used when the target of each replacement is a single character.
When the idea is to rectify erroneous input, this is often called autocorrection. The term is sometimes used even is the feature is mostly used to expand abbreviations. Another term is autocompletion, mostly used when the original text is a prefix of the replacement text.
Emacs calls this abbrevs. Vim doesn't have a more specific name than insert mode mapping. Neither term is likely to be understood by people who don't use the program in question.
If you're writing the documentation for a program that has a similar feature, I'd go for “(automatic) abbreviation expansion” if it's a generic feature, or “autocorrection” or “autocompletion” or “input method” if it has a specific focus.
If you're looking for a generic phrase for Google searches, I think you're out of luck.
Maybe it is Autocomplete.