NSW Trademark almost identical but only with imprint of a company

[MrAndy]

Hello dear Lawanswers.com-Community.

I need your opinion as well as legal advice on the following matter, see below.
Would be very nice for your help, please.

It is about the following situation:
I would like to register a trademark - a word / name to run a homepage and possibly a studio in the gaming industry.
I've used various trademark search tools, including the well-known Uspto-Trademark Website.

- However, with my trademark word, I have not been able to find an already explicit registered trademark.
But only a studio that is also in the gaming industry and has an almost identical name in the imprint.

Hence my questions now:

• Now who has the right to use the word?
• So if I register the Word now, am I on the legal side?
• Shouldn't the studio have registered their name as a trademark to be on the safe side that nobody is allowed to use the brand word?
  On the other hand, most other companies have even secured their company name, i.e. the word and the domain.

For example: Coca-Cola has registered its word as a trademark. If that weren't the case someone could come up and say the brand name is mine.

From a legal point of view, i assume the trademark belongs to the person who registered it first, even if a company, studio or etc. has already used the name but not registered it, it is the fault of the owner. Or have I misunderstood that?

Greetings to you.