I carried myself from gold to platinum last season with Leona and I think the only reason why I find her so strong is because of her ultimate.
The ultimate itself is gamechanging and has lots of uses. As you already said, it can be used for initiating or escaping, for keeping the stun lock up on a target or even for damage and the passive.
When to use it in general depends on the stage of the game (mid/late) and the enemies who you want to stun.
If you are still in the laning phase you usually want to use it aggressively, since you are arguably the most aggressive supporter in the game (Thresh/Lulu maybe aswell but noone has that much burst and CC). If you are versus an adc or squishy supporter that has no gapclosers (Jinx/Varus/Annie/Zyra) you can use your ultimate to land the initial stun on them which (hopefully) guarantees a secure E spell followed by another stun. Although note that if they have flash up, it's really easy to dodge the ultimate since it has a relatively long cast time. An annie tibbers is safely castable because it's almost instant but a leona ult has about .25 - .5 secs delay. On the other side when you are playing versus an ADC with high mobility (Ezreal/Draven/Lucian) you should use your E first and Stun the target with your q after. Just right after the autoattack you should then cast your ultimate to hold him in place long enough to burst him down.
The only time you want to use your ultimate defensively is when you are getting ganked, or you have no chance in fighting the enemy lane.
In teamfights there are a few more options:
- Use it to lock down a high priority target (fed adc for example)
- Use it to help your team getting away
- Use it to protect your ADC/APC
- Use it to Initiate a fight (Be careful with this one)
For all the points above except the first one keep one thing in mind!
Try to hit as many targets as possible!
I've won and lost so many games just because a Leona ult was perfectly placed and hit like 3+ targets of our team. Most comps have another aoe stun or burst and combined with Leonas ultimate it can turn a teamfight in no time even if the team with leona is behind.
But honestly you have to just get a feeling of when to ult yourself. I have a 85% winrate with her but I still can't really explain when to ult or when to not ult without writing a Full guide.
Hope that helped a little and good luck practicing her! Once you can play her it's like free elo ;)