What is the interpretation of the Hadith: "Greeting (salam) precedes speech, so whoever begins to address you with speech (before greeting), do not respond to him," and how should I act if I greet my teacher with salam, and he responds to me with "ahlan" or "marhaban" – should I respond to him or not?

The phrase "Peace before speech" is not an authentic hadith. The first part, "Peace before speech," is weak and fabricated, and the second part, "Whoever begins speaking before giving greetings of peace, do not respond to him," has no basis. Giving greetings of peace (salam) is a Sunnah, and returning it is obligatory. If the teacher does not return the greeting of peace, respectfully and politely explain to him that returning the greeting of peace is by saying "Wa Alaikum Assalam wa Rahmatullah," and there is no harm in then saying "Ahlan wa Marhaban" (welcome).