I think what you've identified is the contrast between collectivism and individualism. The left-right dichotomy is a little more complicated as it is a highly location and history specific differentiation. Various questions deal with issues of what makes up right and left.
I can think of a few examples where your definition might fall apart. I'm American, so my examples may not be perfect for Belgium.
Consider an insular religious community. They all go to church together and have a strong emphasis on caring for their community and consider individual focus to be selfish gluttony or pride. They might vote for right-wing parties as they want freedom to run their local freely-associated community as they wish. They don't want to participate in regulated government services like schools that push different cultural values on their children. They prefer the local church based initiatives that comply with their faith. They still feel strongly compelled to care for others, but prefer the leadership of their church community to the leadership of the broader culture.
Consider a transgender person in a society that doesn't accept the idea of transgender people. That person actually doesn't like the idea of everyone taking care of everyone, as the majority's idea of "taking care of everyone" probably involves persuading or coercing him to adopt a different identity than the one he feels is right. Taking care of someone can mean doing what you think is best for that person despite their protests. Most left wing groups abhor the idea of suppressing people's identities, so their "everyone caring for everyone" is actually tempered by a strong desire for everyone being able to decide what is best for him or herself, an idea pretty close to "everyone caring for him or herself."