Hi everyone I want to thank you for your help and update on how I finally managed to get the VISA extension (as your suggestions):
1-I had a letter from my previous organization (in Songkla) that stated that my period ended on 31st August
2-I went to Songkla immigration office and canceled my VISA previously ending on 8th September, making it ending on 31st August
3-I asked my organisation in BKK to issue the new training course starting date on 31st August
4-I went to immigration in BKK on 31st and got my "under consideration stamp" until I went again today (20th Sept) to get the non immigrant ED EXTENSION stamp
For point number 3, I could have the option to keep the starting day in BKK on 1st September, the only difference was that I would have had to pay 500 baht for 1 day of overstay, but not any other issue.
I hope this experience could help other people who were confused as me.
My passport nation is Italian and the study is an educational training in government hospital. But 31st Aug and 1st Sept are Saturday and Sunday and immigration offices are closed. What do you suggest me to do?