Role of the tank/support

Im just trying to wrap my head around something here, and hoping I can get some input. I understand there is no one specific answer to this, but at least some thoughts would maybe help me in the right direction.

I enjoy playing the tankier supports, my main is Leo at the moment. Laning phase is all well and dandy, but one the laning phase ends I get a little lost or distracted, perhaps. Warding, roaming, helping other lanes. And, of course, teamfights. Its in the team fights I often get lost or lose sight of my role and what I should be doing.
In team fights, should I be at my adc's side the whole time, peeling for them and keeping them safe? Or is it a better option to be the engage, dive in with a well timed ult in the middle of the enemy team to follow up with zenith blade and stun their carries from amidst their lines?
I get the same issue with other champs, like Garen as an example. If I dive the enemy lines for their carries, one of whom is usually the villain, I either get focused and melted, or they ignore me and just chase down my carries, while Im just spinning and hoping to cause chaos. So do I dive, or do I hang out with my team and wait for the enemy team to engage, and just try to help the adc/mid laner by doing whatever I can to keep the enemy champs off the faces of my carries?

Is zoning an enemy carries out as a tank/bruiser more important, or is it more important to just keep your own carries safe by sitting with them?

Obviously a lot comes down to the team comps and what champs are chosen, but I would welcome any basic or vague advice.