I think using the Flag module could be a solution for this. A "Flag" is like a variable, and boolean (it is turned on or off, or set or unset). In this case a flag to indicate either "Visible to vistors" or "Hidden to visitors".
Probably combined with the Rules and the Content Access module. So that when the author SETs the flag to "Visible to vistors" (= The event used in the "Rule"), then via Rules you trigger an "action" to grant access to "the entire node" for anonymous users (if that is what you'd want to do).
If however the author should only be able to specify if visitors can yes or no see the update date of the node (so not to make the entire node yes/no visible), then these refinements/variations apply:
- Refine the meaning of the flag, something like "Update date visible to vistors", versus "Update date hidden for vistors".
- Use the Field Permissions module to grant access to the "update date" fields.
- If flagged as "Update date visible for vistors", then via Rules you trigger an "action" to grant field level access to "update date" for anonymous users also.
If you think like "hm, 3 modules instead of just using 1 extra field?", then think twice ... And note that this approach gives you so much more funtionality. Moreover, modules like Flag and Rules should be part of mostly any site (there are so much usecases for them ...).