The second group you are talking about is easy to categorize:
"...another group. They will start to worship/believe the moment proof of God's existence is provided."
A strong agnostic believes that we can never know whether God exists or not.
A weak agnostic believes that proof of God's existence is possible, we just haven't gotten there yet. This term would cover the second group you are talking about
The first group you describe is kind of hard to categorize:
"nothing will make me worship God - even if God exists - because I don't want to!"
First you have to distinguish belief from worship, they are not the same thing. A traditional Christian believes in Satan but doesn't worship him. So you would have to distinguish two groups, (a) those that don't worship because they don't believe, and (b) those that don't worship even though God exists, because they don't want to.
(a) An atheist believes there is no god or gods.They believe God is a fiction on the same level as Thor, Gandalf or Voldemort, so what's the point of worshipping him?
(b) People who believe God exists, but don't want to worship him. There are some schools of Satanism, known as Theistic Satanism or Reverse Christianity. They believe that God exists, but they prefer to worship Satan rather than God. Closely related are the terms Misotheism, hatred of God (or gods), and Dystheism, the belief that God (gods) is not good, but can be wholly or partially evil.
Beyond these, there are other categories as well, but it would take too long to go through all of them.
To summarize, what you are looking for is not really two categories, but multiple categories, including: Atheists, Strong Agnostics, Weak Agnostics, Theistic Satanists, Misotheists, Dystheists, Ignostics, Nontheists, etc....