Do I have a grace period to overstay my visa in Thailand, or should I do a visa run?

From the stamp in your passport it appears that you may be based in Chiang Mai? If you are and if you only need a further 30 days from the 6th of August the closest border is Mae Sai - cross the border and pay 500 baht (or USD10) for the visa into Myanmar, they will then retain your passport and give a card with a number on it. Have a walk around the market on the Myanmar side a while, then before you exit Myanmar you present the card with the number on it and collect your passport - you should then receive a 30 day Visa Exempt Entry when you enter Thailand (you can get 2 of these per calendar year - you received one when you initially entered Thailand, this will be your second one). Very easy process!

AS the stamps says: must leave the Kingdom within the day specified herein offenders will be prosecuted, so no grace period. What he tried to explain is when you walk in for a new Extension of Stay it will be NOT approved and you get 7 days to leave the country. Best option is to do border run just before or on 6 Aug to get a new Visa Exempt Entry but be aware you can only do this twice a calendar year or visit a Thai Embassy or Consulate nearby and apply for the Single Entry Tourist Visa which gives you 60 days of stay with possible of extension for another 30 days.