I think that book is being a bit dis-informative without providing associated timelines. "An early time" doesn't give any range whatsoever or specify Northern Tai, Central Tai, or Southern Tai/Thai.

The reason this is important is because one of the Tai language groups is in China, Zhuang, in roughly the area considered the origination of the Tai (not Thai) peoples. Zhuang uses a written script called Sawndip, that is based off of the Han character system. AKA (for normal people) Chinese. The Tai didn't become dominant around Sukothai until around 1100 CE.

So, Thai didn't have it's own script until King Ramkhamhaeng in the late 13th century. What got used as written script from 8th to 13th, or more specifically from 1100-1238. Khmer? Chinese characters? Did they just not write? Going back to the geographical origination of the Tai, Chinese characters would make a lot more sense.

Pittayawat Pittayaporn from Chula has some good papers on it if you want to look some more up on it.

Problem with using your textbooks at school is that any school is going to gloss over a lot of the basics, especially linguistics.

edit: Okay I can't not say it. Thailand has also done an amazing job of ensuring that the national narrative is taught, to promote the commonality of "Thai." I'm not disagreeing with the choice to do it, but the actual history of Thailand is so amazingly complex it's almost a disservice that it's so simplified. Then again, I'm American and we beat the Brits by being plucky upstarts and not standing in lines to shoot at the redcoats (sarcasm, American cult of go-it-alone.) Sometimes it's easier to teach kids stuff that's wrong instead of "Well, it's complicated."