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).
Well, when renaming a variable you usually want to avoid touching non-variables so including the $ seems reasonable to me, and I'm used to it, but I see your way of looking at it.