Must I leave the Schengen Area to start a new 6 month period as a Canadian?

My initial answer might be incorrect as of October 18, 2013, see About Schengen 90/180 rule. Under the new rules, the travel you were planning is not possible anymore as you would need to stay out of the Schengen area for three months to be able to get another three-month stay.

Reading the court decision cited in the relevant part of the Handbook, it seems indeed necessary to leave the Schengen area for a few days so as to never stay for more than three months at once. Other than that, your interpretation seems correct.

The relevant bits:

33 Thus, the Commission, and likewise the Finnish Government, observe that a national of a third country not subject to a visa requirement who, having taken care to leave the Schengen Area on the very day of his first entry, has effected a stay of three months less a day at the end of an initial period of six months, could, by leaving that area for a single day at the end of that initial period and entering it again the following day, stay in the area for three additional months during a second period of six months, thus allowing him to move freely within that territory for a period of six consecutive months less a day.

[…]

38 It follows from this that, regardless of compliance with the requirement as to the period of six months following the date of first entry in Article 20(1) of the CISA, nationals of a third country cannot in any event stay in the Schengen Area pursuant to that provision for more than three consecutive months in all. As regards nationals of a third country not subject to a visa requirement, that absolute maximum limit is clear from Article 5(1) of the CISA and Article 1(2) of Regulation No 539/2001.

[…]

42 As for the risk of circumvention of the rules applicable to long stays, as submitted by the Commission, it is sufficient to point out that while Article 20(1) of the CISA allows, as its wording now stands, a national of a third country not subject to a visa requirement to stay in the Schengen Area for a period of almost six months by aggregating two successive non-consecutive stays, it is for the Community legislature to amend that provision, if necessary, if it takes the view that such an aggregation is capable of undermining the rules applicable to stays of more than three months.

Now, don't forget that even if you are not falling foul of the three-month-in-a-semester-rule, your stay must be for certain purposes only and you are not entitled to enter merely because you don't need a visa and haven't overstayed. What I mean by that is that staying for three months and then reentering could invite extra scrutiny about what you are actually doing in the Schengen area.