Catch Error in a SQL Server Logon Trigger

After digging into the document of SQL Server and doing some testing, I feel I can tell you what the problem is and how to fix it:

1. Don't return result in the trigger. The way you try to raise exception is select 10/0, this will raise a exception but not the exception you expect. select 1 has the same effect.
2. Before the LOGON TRIGGER, an implicit transaction has been start, that's why when you want to refuse certain account login, you should use ROLLBACK, it cancel the login process. When a exception has been raised and you were not call ROLLBACK explicit in CATCH block, an implicit ROLLBACK will be called after the trigger executed. As you know, any data modifications made up to the point of ROLLBACK are rolled back. That's the reason your INSERT INTO ERROR_LOG not working.
3. To confirm these points, add a ROLLBACK before the INSERT INTO in CATCH block, and change the value for [ERROR_DATA] to ERROR_MESSAGE(), you will get a exception message in the ERROE_LOG table.

Code below is base on your requiement, but the requirement that "in case of whatever error the login is allowed to login and write a error log" I cannot find a way to implement yet. Keeping work on it. And this code cannot be tested in my computer, so it could have incorrect things, you have to fix them by yourself.

ALTER TRIGGER [LOG_TRG_02]
ON ALL SERVER WITH EXECUTE AS 'SA'
FOR LOGON
AS
BEGIN
	BEGIN TRY
		IF ORIGINAL_LOGIN() = 'OMEGACAEVDEV1'
		BEGIN
		-- not allowed to logon
			ROLLBACK;
		-- Insert Trail of OMEGACAEVDEV1
			EXEC [OmegaCoreAudit].[OMEGACA].[P_ACC_UNF_TRAIL]
			0, 'trail_details', 'ip', 'server', 'db', 0 ;
		END;

		IF ORIGINAL_LOGIN() = 'OMEGACAEVDEV2'
		BEGIN
		-- Allow login and Insert Trail of OMEGACAEVDEV2
			EXEC [OmegaCoreAudit].[OMEGACA].[P_ACC_UNF_TRAIL]
			0, 'trail_details', 'ip', 'server', 'db', 0 ;
		END;
	END TRY
	BEGIN CATCH
		ROLLBACK; --refuse logon with exception, and make the insert statement atfer this work.
		          --I cannot find a way to allow login with exception yet.
		INSERT INTO OmegaCoreAudit.OMEGACA.SYS_ERROR_LOG 
		([LOGIN_USER_NAME], [DB_USER_NAME], [USERHOST], [IP_ADDRESS], [OS_USER], [BG_JOB_ID], [ERROR_DATA])
		VALUES
		(ORIGINAL_LOGIN(), 'USER_NAME', 'USER_HOST', 'IP', NULL, NULL, ERROR_MESSAGE()); 
	END CATCH
END 
GO

There is a code:

-- not allowed to logon

rollback;

It looks like you are using implicit transaction. It could be a reason. You can try to add BEGIN TRANSACTION before Insert ERROR_LOG, and  COMMIT TRANSACTION after insert.