ALTER TRIGGER [DEFINITIONS].[CLASSIFICATIONS_INS] ON [DEFINITIONS].[CLASSIFICATIONS]
FOR UPDATE
AS
BEGIN
--IF TRIGGER_NESTLEVEL() > 1
-- RETURN
UPDATE DEFINITIONS.CLASSIFICATIONS
SET UPDATE_DATE=GETDATE(), UPDATE_USER=DIARY.fn_CURRENT_USER(@@SPID),UPDATE_TERMINAL=HOST_NAME()
FROM DEFINITIONS.CLASSIFICATIONS C
INNER JOIN INSERTED I ON C.CLASSIFICATION_ID=I.CLASSIFICATION_ID
END


ALTER TRIGGER [DEFINITIONS].[CLASSIFICATIONS_UPD] ON [DEFINITIONS].[CLASSIFICATIONS]
FOR UPDATE
AS
BEGIN
--IF TRIGGER_NESTLEVEL() > 1
-- RETURN
UPDATE DEFINITIONS.CLASSIFICATIONS
SET UPDATE_DATE=GETDATE(), UPDATE_USER=DIARY.fn_CURRENT_USER(@@SPID),UPDATE_TERMINAL=HOST_NAME()
FROM DEFINITIONS.CLASSIFICATIONS C
INNER JOIN INSERTED I ON C.CLASSIFICATION_ID=I.CLASSIFICATION_ID
END

After diagnose I found the error in the bold text.
I change it for the insert trigger FOR INSERT and solved the problem