As mentioned, « for caring me » could be poor English if it means « to look after/to take care of ». Please note that grammatically speaking, « for caring me » is correct since « caring » is used as a gerund in this context, and the prepositional phrase « for caring me » means « to nurse/to treat ». When it is used to mean « to nurse/to treat », it is OK to say « for caring me » since the meaning implies: « for treating/nursing my wound(s) ». Out of context, it might sound odd because this is likely to be used specifically in medical contexts, but it is correct. However, saying « for caring me » to express « to look after/to take care of » without implying that someone is physically hurt is not really something that a native English speaker would say. Knowing that, « for looking after me/for taking care of me » suits best. « for caring about me » means to give attention, but not necessarily « to look after/to take care of/to nurse/to treat ». The context and what one wants to express are important. I hope that helps.

The phrase "Caring me" is poor english. You can "care about someone" meaning you have nice thoughts about them. You can care for someone meaning you love or like them. And you can take care of someone meaning that you look after them/help them

care for - (1) have feelings for or (2) take care of care about - like empathy, you would feel distress if they feel distress . also "care for": "like", eg Chocolate is not something I care for

The second one is correct English.