Is it possible to quit a TVL13 job in probation period? I would like to ask whether it is possible to quit a wissenschaftlicher mitarbeiter job based on a TVL13 contract in the middle of probation period? Or do I have to wait until I finish it?
I have found the law corresponding to the different notice periods after the probation period is over, but I am not sure what is allowed before it is over. If I am not wrong, the employer can fire me without notice during that period.
In this website, I found the following:

until the trial period expires    2 weeks to the end of the month

Is this information reliable? It seems to be referencing the same law, so I'd say yes. But, I would like to double-check.
If you want legal advise, then you need to talk to a lawyer or your union representative. It is a bad idea to trust random people on the internet on such issues.
My experience in Germany is that if you want to leave you don't go through that process. Instead you talk to your supervisor. (S)he will express regret and want to know why, but in the end (s)he will not want to keep someone who wants to leave. So together (with the HR department) you will make a contract that terminates the existing contract at a convenient time for both regardless of the notice periods.