What is the meaning of "self-conscious"?

Question

What is the meaning of "Self-conscious"?

I know self-conscious means worrying about how other people think about you. For example, I'm very self-conscious about my appearance. But, in the next following example, I cannot how people are self-conscious in that kind of context. Can you help me?

When we hear that someone we care about has lost a job, started chemo, or is going through a divorce, our first impulse is usually “I should reach out.” Then right after that impulse doubts often flood our mind. “What if I say the wrong thing?” “What if talking about it makes her feel self-conscious?” “What if I’m overstepping?” Once raised, these doubts are followed by excuses like “He has so many friends and we’re not that close.” Or “She must be so busy. I don’t want to bother her.” We put off calling or offering help until we feel guilty that we didn’t do it sooner…and then it feels too late."

Answer 1

This is definition 2 from Merriam-Webster

uncomfortably conscious of oneself as an object of the observation of others

or definition 1 from Lexico

Feeling undue awareness of oneself, one's appearance, or one's actions.

Your example about your appearance is simply a specific example of this, but there can be many reasons why you can feel self-conscious. In the quoted article, reminding someone that you're aware that they're going through something unfortunate can also do it. This results in a conundrum: you'd like to comfort them, but raising the issue can cause discomfort.