Yes: the employer should notify Thai Immigration and Labor departments due to a change in circumstances. In addition, you should notify the Immigration; therefore, affecting your status.
If neither party notifies then the prior status shall continue indefinitely.
The Thai government is serious about foreigners working illegally (ie without appropriate visa & work permit) in Thailand; whether from their hotel room, in a market, "farang" or sister-Asian.
You seem to think that the Thai authorities consider a foreigner in a coffee-shop exempt from the Thai law whereas a factory-worker in Chonburi is not. The Thai Immigration is losing money when nomads who do not have a LTR, SMART etc... Eg: yeterday 5 people were caught working on ELITE visas and taken to Lumpini Police for illegal labor. In your opinion, it seems the Thai authorities differentiate between IT criminals with a hardrive and Property owner criminals with a hard-hat.