My JSON data contains dates in one of three formats:

YYYY
YYYY-MM
YYYY-MM-DD

The field is: Items::Initial_Release

Instead of the text, I would like to have it display that as a proper date based on what's there.

1984:               Year Only
1984-08:         August, 1984
1984-08-08:   08 August, 1984

Based on the available options for formatting the date, I gather that the FileMaker Date format probably can't deal with less than the full amount of data? Or even changing it, having it be Day first if there is full data, otherwise Year, or Year, Month only.

Otherwise, it looks like I would have to have the date field in three different formats and use Hide If .. the data is 4 characters, 7 characters, or 10 characters?

As well, no matter what I do with those format settings above, I can't enter anything less than 10 characters in the field if it's defined as a date field.

I'd like to be able to get things by date in queries, so keeping it as date format would seem to be the better way to make that work.

Only date fields can be formatted as dates - and a date field must contain a full date (i.e. a specific day in a specific month in a specific year) in order to be valid. 

There are several ways to handle partial dates (e.g. separate fields for day, month and year). In your example, it seems the simplest way would be to keep the original "date" in a Text field, and make your own calculation field to display it the way you want.