Photo 1: Note that the fighter's elbow has travelled past the centreline (as roughly indicated by his nose). He rotates his foot beyond neutral.
Photo 2: The elbow has not yet passed the centreline. His foot remains neutral.
You can try this now, at home, and feel how twisting the front foot inward facilitates the increased follow-through achieved by the fighter in the first photo. This is typically desirable when you are intending to strike through the target and wish to commit more fully to the technique with the intent of maximising damage.
Keeping the rear foot neutral may be sufficient if you are merely intending to throw a quick elbow designed to stun and/or distract, and if you need to maximise retraction speed for defence or a follow-up technique.
Bringing the tip of the thumb of your striking arm into contact with your opposite shoulder (at or just after the moment of impact) provides a good cue; a reminder to bring your elbow further around than you otherwise might.
In relation to the question of whether or not to keep the wrist relaxed, the whole shoulder and arm should remain sufficiently free of tension to enable you to create a snappy, almost whip like action by collapsing the elbow into the strike. You may however wish to tense at the precise moment of impact, just as you would with a punch. Experiment on a bag or pads.
This article and video provides some excellent further information as to the biomechanics of elbow techniques, and in relation to some common mistakes and supplemental training ideas.