Depends how you look at it - to me it's more like `$variable_name` than "variable name starting with $" so selection of `variable_name` only doesn't seem like a too special case (also in some cases like when you e.g. want to rename a variable, you want to keep the first `$` and I think this is something users are used to).
By the way is there any need for `editor_find_current_word_sciwc()`? I think it uses some hard-coded Scintilla word boundary rules and it just is another source of possible source of word boundary inconsistencies.