How did the Buddha debate? Was he trying to convince or persuade people?  Did he go towards them or did he let them come to spread the Dhamma?  How did he react when he was not understood or listened?

I have noticed that I often tend to get carried away in debates where I try to convince my interlocutor of the truth of Buddhism and I get angry when he doesn't listen to me or disagree with me and then I blame myself for having debated in the first place.  Then I often find myself with the feeling that I should have kept quiet, and I remember this quote:


  He detested objective truths, the burden of argument, sustained
  reasoning. He disliked demonstrating, he wanted to convince no one.
  Others are a dialectician’s invention.
  
  
  Cioran, The Trouble with Being Born
  


Should we keep quiet in a "save yourself before saving others" logic?
The Pali suttas say the Dhamma is taught only to those who are interested, per AN 9.5:


  The best sort of kindly speech is to teach the Dhamma again and again
  to someone who is engaged and who lends an ear.
  
  Etadaggaṃ, bhikkhave, peyyavajjānaṃ yadidaṃ atthikassa ohitasotassa
  punappunaṃ dhammaṃ deseti.
  
  https://suttacentral.net/an9.5/en/sujato


The Pali suttas say debate is not the purpose of Dhamma, per MN 22:


  They just memorize the teaching for the sake of finding fault and
  winning debates. They don’t realize the goal for which they memorized
  them. Because they’re wrongly grasped, those teachings lead to their
  lasting harm and suffering. Why is that? Because of their wrong grasp
  of the teachings.
  
  https://suttacentral.net/mn22/en/sujato


The Pali suttas generally show the Buddha debated people who visited him to debate him, such as MN 35: 


  Now at that time Saccaka, the son of Jain parents, was staying in
  Vesālī. He was a debater and clever speaker regarded as holy by many
  people. He was telling a crowd in Vesālī: “If I was to take them on in
  debate, I don’t see any ascetic or brahmin—leader of an order or a
  community, or the teacher of a community, even one who claims to be a
  perfected one, a fully awakened Buddha—who would not shake and rock
  and tremble, sweating from the armpits".
  
  Hopefully, some time or other I’ll get to meet Master Gotama, and we
  can have a discussion. And hopefully I can dissuade him from his
  harmful misconceptions.”
  
  https://suttacentral.net/mn35/en/sujato