Actually the right side is correct in the original diagram for Japanese.
But in Chinese the stroke order of the right side is actually as you said 6,8,7,9.
However the left side is incorrect whether Chinese or Japanese.
I think there are some cases where the stroke order for some kanji is not strictly defined, but only for some kanji that are not in the Joyo list such as 墟.
I’m seeing the following three! possibilities for 墟.
Not quite. I don’t know what stroke order is like in Japanese calligraphic circles – maybe everyone uses exactly the same stroke order no matter what – but in Chinese calligraphy, stroke order changes depending on what form you’re using. For example,
This is our stroke order for 王 in 楷書. However,
The Japanese stroke order is what we use in 行書, because it’s more fluid and helps us to to create a character that looks more centred in the process without accidentally leaving asymmetrical traces of ink when we go faster. The truth is that all this stuff is about the stroke order of 土, which Chinese changes in 行書 when it’s at the bottom of a kanji component: writing from top to bottom, the standard order is horizontal, vertical, horizontal; the 行書 order is vertical, horizontal, horizontal. Even if you go fast without lifting your pen/brush, the 行書 order will simply add two threads of ink that form a V, which is still fairly aesthetically pleasing and reduces the need to pay special attention to stroke ends. (For an example of what I mean ‘by special attention’, consider the standard stroke order: if you go fast, you’ll have to find a way to allow the vertical stroke to segue into the final horizontal stroke on the bottom, ideally without letting it curve too much off to the left.) On the other hand, however, in Chinese calligraphy, 土 is written using the standard stroke order even in 行書 if it’s at the top of a kanji part.
Similarly, for 木 when it’s in what I call ‘radical form’ in a reduced form on the left side of a kanji (English doesn’t have a proper equivalent for 偏旁, you see), there are three stroke orders. I’m pretty sure Japanese calligraphers know all three as well:
楷書: horizontal, vertical, leftward curve, dot
行書: horizontal, vertical, leftward curve, sharp rightward upward backtracking to link to the right half (the dot is omitted for speed)
草書: vertical, horizontal, leftward curve, sharp rightward upward backtracking (basically, the end result is sort of like ‘vertical + Z’)
As such, the Jitenon stroke order is actually the 草書 order, but with an extra dot. Of course, perhaps Japanese 草書 is different, but since 草書 is a form with its own set of abbreviations, and its objective is speed (it was created for taking notes in the imperial court or on the battlefield), I really doubt anyone would bother with the standard stroke order, which is slower.
Just for your information, from what I’ve seen, Japanese strike order is often (but not always) the same as Chinese 行書 stroke order.
It’s really up to you, but if you want my opinion, I’d go with the one from kakijun.jp because it’s closer to the stroke order for the simplified form that you posted. That aside, kanji are mostly written left to right and top to bottom, with vertical strokes being the main exception to the left to right rule – they can sometimes be written first – so the versions that construct a mirror image from right to left seem wrongheaded to me. If you can find a source with the official stroke order though, and it turns out that’s the one, then by all means, stick with it.