Why does 「右」 start with vertical stroke and 「石」 starts with horizontal one? What kanji-writing rule prescribes it?
In terms of Chinese calligraphy order,
The order that「𠂇」is written depends on the object underneath it. If the component underneath it doesn't exceed the horizontal stroke of「𠂇」(e.g. 右、有、布), then write the vertical stroke first; otherwise (e.g. 友、左) write the horizontal stroke first.
「石」is a different case; it is a top-down construction, so the top part (the horizontal stroke) goes first.
For「𠂇」, this is due to balance reasons; if the object underneath「𠂇」is narrower than the horizontal stroke of「𠂇」, then finishing「𠂇」with the horizontal stroke last guides the limits of the object underneath it.
I don't know if @istrasci's answer in the other question is something that Japanese reference books explain it as, but we do not explain it that way in Chinese calligraphy. Cases where that explanation doesn't work can be seen in characters like「尤」and「厷」(which is the left side of「雄」).
Bottom component narrower than horizontal stroke of「𠂇」
Bottom component wider than horizontal stroke of「𠂇」