the difference is:
Answer 1
"It is important for me to..." means that what is important is that the speaker does the action, and
"It is important to me that..." means that the speaker considers it important that something happens, which might be an action by someone else (e.g. that you arrive on time), but could be something else (e.g. that the speaker arrives on time, or that the weather is sunny, or anything else).
Answer 2
Yes, I would say "It's important to me that you arrive on time" means personally you want that person to be there, versus "it's important for me that you arrive on time" would be appropriate
if it was like you two were both going to meet with some investors for your company, so their opinion of the individual speaking will be affected by the second person showing up on time or not. Like it's more important in an external way rather than just personally internally caring. But since you normally personally care even in this situation, "to me" is still right and I would say preferred. I wouldn't use "for me" in your situation unless I really needed to stress that other people would think less of me and like, my job is on the line.