Welcome to our community and thank you for your question!
In general you are allowed to travel if you are an acknowledged refugee. You got German travel documents (blue passport aka “Konventionspass”) that enable you to travel to a lot of countries (for further information please check out this thread on the Wefugees community: https://www.wefugees.de/k/questions/2024
However, in any case you are not allowed to travel to your home country. The German state provided you with protection based on the assumption that your home country is either not willing or not able to provide it for you. If you go there, they will draw the conclusion that you are not in need of refugee protection anymore and withdraw it as soon as you try to re-enter Germany.
Unfortunately I can’t tell you which passport they will stamp. May I ask why it plays a role in this case though? And why are you afraid that Lebanese officials won’t let you leave the country with your German documents? I’m just trying to grasp the situation better ;)