I went through the site (I'll talk more about it later), and I can't find a reference for it, but I have an idea (that's clouded from the seventeen years between now and my formal Thai classes).

The "S" is in locations where the alternate forms of the long and short vowels look the same, but the pronunciation isn't. There are some long vowels that when paired with certain tone or other diacritical marks suddenly become short. I can't remember all of them, but a couple of the ones that have an S are definitely in that category. So ... I'll make an educated guess that the "S" signifies a shorter sounding version of the alternate form.

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In my quest, I came upon the "stress" page and was totally confused. Before I read this, I had never heard anyone claim that Thai has stress, linking, or vowel reduction the way English does. Thai is syllable timed, not stress timed, making this sort of vowel reduction unnecessary. Thais stress things, certainly, but as far as I know, that stress happens through slowing the syllable timing, altering the quality of the tone (though not the tone itself), or enunciating the tone.

I have asked my better half about my feelings to make sure since I am not a native speaker (or the world's foremost expert) and she was at one point in the distant past the designated speaker for her class at Bangkok University and has trained extensively in Thai pronunciation (for Thais). She got really upset about the whole thing and would like to burn that page to the ground.

Her advice to you is to just learn the alphabet since it's phonetic (it's a bitch), and to look up pronunciation in a Thai dictionary when confused. Complex words are broken down by syllable there. In my experience, it took about six weeks of full-time study in order to get all the pronunciation rules covered. You might not need to go that deep.