If you add "see you later" it is like saying that the conversation is on hold and it can continue. This is a fast polite ending because it is not an ending. See you later

In general,people often say"good bye,take care" when their meeting is in the end and show that your friends should take care of themselves cos you care. To me,almost i have never said the words"Bye/good bye" with whom i care;because in my own opinion/point when i say that it means i'll go away and never see them again(~,~)So i always just say"All the best.Take care,cheers:)"

You are correct

No that's not mean at all.

Well... That's not what I think when someone says that to me. Plus, the fact you're saying "Take care" implies you want your friend to take care of him/herself, so it means you care. It depends on the person but I highly doubt he/she will think you are cruel or don't care.