To build on @Databaser's suggestion try this:
DATETIME_FORMAT([your date field], "dd")
This returns the following abbreviations:
The issue you are encountering is because you need to pass the formula one of the supported specifiers listed on this page: https://support.airtable.com/docs/supported-format-specifiers-for-datetime-format
If you need just the first letter of the abbreviation except on tuesdays/thursdays you could write:
I hope this helps!