Calendar appointments are a good example where a timestamp without a timezone (java's Instant) is appropriate. But a birthday doesn't have a time, so you would want to use LocalDate (a date without a time and timezone) for that.

In 99% of the cases you only really want to involve timezones when presenting something to the user or taking input from them.

So you would create your calendar event by entering "9am, timezone X", and the app would most probably just apply that offset and store that as a timestamp. But if it's an all-day event... I should probably take a look at android's calendar content provider.