Return Bit Value as 1/0 and NOT True/False in SQL Server

Try with this script, maybe will be useful:

SELECT CAST('TRUE' as bit) -- RETURN 1
SELECT CAST('FALSE' as bit) --RETURN 0

Anyway I always would use a value of 1 or 0 (not TRUE or FALSE). Following your example, the update script would be:

Update Table Set BitField=CAST('TRUE' as bit) Where ID=1