Are you suggesting that all people who get tattoos have some underlying emotional distress or suffering, and therefore is self harm?

No, many do that for those reasons, some people just copy what others do.(Herd mentality)
What determine self-harm is the net negative outcome on the individual and society unless it’s self-sacrifice.

The rule here is if it bad for the individual, it’s bad for society unless self-sacrifice.

Self-sacrifice might be perceived as bad for the individual but it’s a net positive for society.