I also noticed that the Rumba on the site's instruction has the left foot going forward to begin whereas I learned that the left foot goes to the left, replace right, left forward.
I think you mean LF side, close RF to LF, then LF forward.
You'll notice that it's actually the same step, but with a different starting point. A box is a box, whether you begin with the forward step or the side step. Apparently, the Fred Astaire studio invented this alternate version of the box in the early days to avoid copyright issues with Arthur Murray. Or so I'm told.
To dance the box step starting with the side step (QQS) rather than the forward step (SQQ) is simply a matter of preference. Typically, people tend to do whichever version they learned first. I've noticed that the QQS rhythm fits more neatly into a written syllabus. But from a musicality standpoint, SQQ is better suited to the bass line and percussion of most Rumba music.
I taught for Arthur Murrays for about 4 years, then for an ex-Fred Astaire studio for 5 more. So I've had sufficient experience with both ways of dancing the box. When I went independent I ultimately went with what I believe is the more musical of the two methods, SQQ. Sometimes I still begin with the side step, but to stay true to the music, I'll begin on count 3. That is what I recommend to anyone who has the choice (ie unencumbered by chain studio mentality and syllabus restrictions).
Regards,
Jonathan Atkinson