Since you didn't specify a country, I'll answer on the basis of the UK, given that it is a major financial center. Just bear in mind that exact definitions may vary in other jurisdictions.
It is a common misconception that "public company" has the same meaning as "listed company". Actually the term "public company" does not mean what you think it does. Under section 4 of the Companies Act 2006, a company can be either a private company or a public company. Being a public company is simply a matter of how your company is registered, and does not imply that your shares are listed on an exchange. That is an entirely separate procedure, governed by separate rules, and such a company is typically called a quoted company, traded company, or listed company (albeit technically the term listed only applies to the Main Market of the London Stock Exchange, but this distinction is not commonly made). You may also see the word public prepended so e.g. "public listed company". The important thing to bear in mind is that "public company" does not mean the company is listed. In fact there are far more unlisted PLCs (Public Limited Companies) than there are listed ones.
"I also assume (but not sure) that in case if they are offering 100% of stocks to the public then they want to be private company"
If a company has offered 100% of its shares to the public then it is very unlikely to be a private company (as defined in the Companies Act). This is because section 755 of the Companies Act 2006 prohibits private companies from making an "offer to the public any securities of the company".
When company decides to do IPO, does that mean they are already a public company OR does it mean they are becoming (or want to become) public company? I'm asking if company that does IPO can already be public one before IPO is finalized (or even started)
Yes, because of the prohibition on private companies offering shares to the public, an IPO cannot take place without the company being a public company. This means that they will either already have been a public company before listing, or will convert to a public company during the listing process.
If instead you meant to ask whether a company doing an IPO can already be a listed company, then the answer is no. IPO stands for Initial Public Offering. Thus the term only applies to the first time a company lists its shares. Subsequent offerings are known as secondary issues.