SQL Server Deadlock on two updates due to index lock order
The optimal index for those two queries is not far from the existing definition of the IK_HTT_ACTION_LOG_1 index (add ACTION_UUID as an INCLUDE to the improved index below):

CREATE INDEX nc1
ON dbo.HTT_ACTION_LOG
(
    TRANSITION_UUID,
    STATUS,
    ACTION_SEQ
);


The first query is:

UPDATE dbo.HTT_ACTION_LOG
SET [STATUS] = 'ABORTED', 
    CLOSED = GETUTCDATE()
WHERE
    TRANSITION_UUID = '{F53ADDDA-E46B-4726-66D8-D7B640B66597}'
    AND [STATUS] = 'OPEN';


Giving the following execution plan:



The second query can be expressed this way:

UPDATE ToUpdate 
SET [STATUS] = 'RUNNING {36082BCD-EB52-4358-E3D3-4D96FD5B9F0F} 1360094342'
FROM
(
    SELECT TOP (1)
        hal.[STATUS]
    FROM dbo.HTT_ACTION_LOG AS hal
    WHERE
        hal.transition_uuid = '{F53ADDDA-E46B-4726-66D8-D7B640B66597}'
        AND hal.[STATUS] = 'OPEN'
    ORDER BY
        hal.ACTION_SEQ ASC
) AS ToUpdate;


Giving this execution plan:



Both queries now access the same resources in the same order, while locking many fewer rows on the read side of the plan. The execution engine will automatically take UPDLOCKs when reading the new index, providing the serialization you are looking for.