How to care by not caring?

…………… Or how to not care by caring?

It's pretty obvious that I am using the word “care” in 2 different senses but it's care nevertheless. So what are these 2 different types of care? I will just give 2 examples and it might be clear after that.

1. Suppose you have a 1 year old kid and you have taken him/her for vaccination. You know the kid is going to cry a lot due to the pain of the injection. Will you stop vaccination because of that or will you go ahead? Here you have cared (By careful considering of the importance of vaccination) for the kid by not caring (about the pain the kid has to suffer).
   
   
2. Suppose you have a strong ideology and you have your reasons for it. The ideology is a consequence of your experiences and learnings. Now that ideology might not particularly please the close people around you. Also, it is very difficult for them to understand the importance of the ideology because they have not had the experience which you have had. So it is very difficult for them to put themselves in your shoes. But it troubles them nevertheless. So what are you going to do about it? If you give up on those beliefs because you “care” for them, somewhere you have not cared by caring. I would say not caring because it gives a distorted image of what you are to them but probably they deserve more. They might deserve to know the sensitivities behind your ideology. It's just that you need (also them) perseverance to finally get to a point where the ideology stays and also the relationship. And that is real caring in my opinion.
   
   
   
   
   So yeah. In my title, the first “care” is giving importance to the person by giving importance to the deeper meaning/purpose behind certain beliefs and actions. The second “care” is giving importance to the person by giving importance to their emotions of pain, trouble, fear etc etc.