Hi Irene,
When I wrote The way this is punctuated suggests to me that Mr. Jones who teaches Chemistry wasn't as athletically successful as Mr. Jones who teaches History I was trying to tell you, in a brief and slightly amusing way, that it is correct punctuation, but that it has the kind of meaning I suggested.
In other words, in ...Mr. Jones, who teaches Chemistry, ... the commas make who teaches Chemistry just optional extra information about Mr. Jones.
But in ...Mr. Jones who teaches Chemistry ... the lack of commas means that who teaches Chemistry is needed to identify Mr. Jones. In other words, it suggests that there are two teachers called Mr. Jones and only one of them teaches Chemistry.
So, both versions are correct. It just depends on what you mean.
I'm sorry if you didn't understand or appreciate my attempt at humour. Some people don't think I'm funny. On the other hand, I notice that Davkett did a lot of work to give you complete answers, and you don't seem to have had time to thank him yet. I think these small human gestures, like politeness and even weak humour, are pleasant and important, don't you?
Best wishes, and please write again if you need any more help,
Clive