It's a normal practice for the intern in Bangkok when do the visa extension with the immigration. The ministry of Labour will indicate on your work permit that you are an intern.
You'd better check the required doc for the non-ed visa extension with the immigration(assuming the one at Chaeng Wattana ) I 've learnt you will need the certification letter supporting your visa extension from your embassy/consulate in TH.
I have learnt that the officer at the counter for foreigner, Building no.4 at the Land transportation office close to BTS Mochit is the right person to you the correct answer. You can contact them on working them during 8:00-12:00, 13:00 - 15:00. Bring your Vietnam driving license otherwise they can't check it for you
if it 's done by your landlord then suggest you keep the copy of your approved TM.30 (paper) & a print out of the TM.30 submission caption via the mobile application. Keep it in your passport just in case.