Its slightly a translation thing and slightly a culture thing. Direct translation can be "My teacher/instructor" since often unless absolutely necessary korean doesnt often use pronouns. Also since in their culture job title cones before name usually even if the teacher isnt your own. At least thats what ive read/heard
There is zero percent chance it's this. Ideum has no reason to lie as he doesnt even know the teacher is Jaeyis brother. The switching of names was meant to show that Ideum thinks of the teacher as a friend ergo he referred to him casually by his name rather than his title (supported by the fact he explains after how they have a lot in common) he switches to calling him his teacher in order to give context to Jaeyi bc he doesnt know it's actually his brother to him hes just another teacher so thats how he explains who he met with.
Np so him slipping up is bc he caught himself sounding impolite when talking abt a teacher bc he doesnt know theyre brothers, even tho the brother himself insisted he could be casual. I dont think he is trying to hide anything, I think he is trying to avoid sharing to much of his personal life since they agreed theyre just fuck buddies.

I don't understand why ideum keeps referring to him as my instructor when he's not his instructor his instructor is the friend of ml. Is he lying . Is it a translation thing