Advantage of raising rights issue over public offering are:
Lower cost issue
Maintains existing shareholding.

My doubt is this:
In private co, the existing shareholding looks significant but is it really relevant for public companies.

This makes little sense as written. Try again.
'Rights issue' means 'lower cost' and 'maintains existing shareholding?' Define that?
What do you mean, 'really relevant?' That means nothing useful.
You are trying to compare private shares vs public?

If you are going to be a student of finance, you need to try to be more specific.
You can't just read a book and then condense it to 50 words.

I recently participated in a rights issue, in this case the company was struggling with market relevance, its share value having bottomed out and they wanted to pivot to a new market, so bringling existing shareholders with them was important otherwise they would have diluted their equity further.

This company would not have been able to go to the market with confidence