OP: If they say that in Buddhism there is no punishment (like in the monotheistic religions) but in the same time they belief and teach
rebirth, then there is punishment and therefore they are contradicting
themselves that no punishment follows. It's not an entity that causes
the punishment, but nonetheless punishment.
Firstly, there is no punishment or "cosmic justice" system in Buddhism.
Rather, it's cause and effect. If you knock your head on the wall, you will feel physical pain. That's cause and effect. Not punishment.
Even in your current lifetime, you can experience cause and effect according to the Sigalovada Sutta:
Bad friends, bad companions,
Bad practices — spending time in evil ways,
By these, one brings oneself to ruin,
In this world and the next.
So, you don't have to wait till the "next world", bad companions and bad practices can affect you in this life itself.
The same sutta explains the effects caused by intoxicating substances:
"These are the six dangers inherent in heedlessness caused by
intoxication: loss of immediate wealth, increased quarreling,
susceptibility to illness, disrepute, indecent exposure, and weakened
The same sutta explains the purpose of the other four precepts both for this world and the next:
The Buddha said this:
"Young man, by abandoning the four impure actions, a noble disciple
refrains from harmful deeds rooted in four causes and avoids the six
ways of squandering wealth. So, these fourteen harmful things are
removed. The noble disciple, now with the six directions protected,
has entered upon a path for conquering both worlds, firmly grounded in
this world and the next. At the dissolution of the body after death, a
good rebirth occurs in a heavenly world.
"What four impure actions are abandoned? The harming of living beings
is an impure action, taking what is not given is an impure action,
sexual misconduct is an impure action, and false speech is an impure
action. These four are abandoned."
For lay persons and others, the Buddha prescribed the minimum set of training rules, which is the five precepts. These are not commandments or prohibitions. Rather, they are training rules. It is up to you whether to undertake the prescribed training or not.
For example, you may not be forced to join the army. But if you join the army voluntarily, you would have to observe the army's training rules. There's a purpose to those training rules, towards achieving the goals of the army.
Similarly, you are not forced to practise Buddhism. If you choose to do so, the minimum training rules would be the five precepts, towards achieving the goals of Buddhism.
Whether you practise the five precepts or not, you are still subject to cause and effect. No matter what you choose to believe or practise, there is still cause and effect in this world and in this life.
To me, the phrase "at the dissolution of the body after death, a good rebirth occurs in a heavenly world" clearly refers to rebirth. But as a secular Buddhist or even non-Buddhist, you may interpret it as a metaphorical mental reappearance from moment to moment, into another state of mind.
Regardless of how you interpret that, cause and effect definitely affects you in this life itself. It's up to you to choose to accept or reject the voluntary training rules of the five precepts, towards the Buddhist goal of ending suffering.