The reason you hear so many different things is because all of those are important, and a support does all of them depending on the situation. So, in order to be a great support, you have to be first great at assessing situations.
For instance, a super super useful trick is keeping track of how your enemy laners move. When they suddenly change their pattern, you know something has changed right? Generally this means something, did they move back? Forward? Both could mean their jungle is coming- they want to let you push, or the jungle is almost there and they want to tangle you up. Simply seeing a difference in behavior can trigger you to gtfo. Assessing situations.
This carries over into nearly everything. Can we all-in our lane? Can we trade our lane? Can we trade them down and then all-in? Can I deep ward their blue, can I roam mid to gank. All of these things come from assessing the situation.
To more specifically address your question:
Supports should have less income, because your income is going into the carry.
A support should be more situationally aware of where jungle is, if top has tp, and if mid is in lane. You should ward more. This has a few varied benefits:
- When you leave lane you lose exp, but your adc gains exp.
- The enemy will likely push in so that your jungle can come gank.
- If you're not in lane, the enemy has to wonder where you are. Are you ganking mid? Warding their jungle? Baiting your enemy laners to push the lane to gank them with your jungle? You're creating questions for the enemy to spend resources thinking about.
A support should be the most map aware and ping the most in my opinion. You don't have to cs, so all of that attention should be spent else where. The better you get the easier it will become to do all of this at once.
Being attached to your carry, thats a tricky one:
Back to assessing your situation. If you leave, can your carry 2v1? Can your carry easily cs under tower? Is he good enough to cs under tower and pressure or will you frustrate him and he won't understand why you left? Can they dive your carry under tower? Where is their jungle? How much time will you have before your carry is forced under tower? Does the enemy have the lane frozen out on him so he can't get exp?
All of these questions help you arrive at the answer. The basic question is, will you gain more or lose more by leaving lane? You will lose exp, your adc will gain it. Will you be right back? If you drop a deep ward, will the jungle even be there any time soon, where was he last? Is everyone else where they belong on the map so you won't get caught? If you get caught by the enemy jungle can you get away? If you screw up is it worth your flash or will you need it to engage a fight?
If you want to leave your adc, first, be sure hes strong enough to leave- and good enough to leave. If he/she/it is fed and goes ham and just dies, its a huge waste. Leave he/she/it warded up. If they are pushing a lane out alone, leave them wards if you need to be with your team for a seige/split senario. If you can't leave your adc safe, don't leave. If your team isn't safe because you and your carry aren't there, ping them back or set the wave to push and goto them.
All in all there is no this is what you do all the time. You assess and adjust to the situation. You may make a decision and the see the enemy jungle on a ward headed toward your carry- well, its time to change your mind and go back to your carry.
Assess and react. The best supports will do all of those things you mentioned exactly when appropriate for the situation.