As per this English SE question, "in ordinary" can mean that it is someone who works regularly for the monarchy. To use their Wikipedia quote:
In relation particularly to the staff of the British Royal Household, and more generally to those employed by the Crown, ["in ordinary"] is used as a suffix showing that the appointment is to the regular staff, for example a priest or chaplain-in-ordinary, or a physician-in-ordinary, being a cleric or doctor in regular attendance.
I'll admit that I'm really not certain on the "magicians-canonical", especially if used in contrast to "magicians-in-ordinary". However, "canonical" is sometimes used to distinguish religious jobs versus secular ones ones as per the term canon to refer to a priest living in a religious order.
This might be supported that "Non-stipendiary" is itself often a religious term (although not always, non-stipend positions also being present in areas such as universities and theatre), referring to priests and ministers who are self-supporting in the Church of England. As they do not receive a stipend for their services, they are supported by either donations or another job.
All in all, with all three items combined, I suspect that Ms. Clarke is making an allusion to the Church of England and how it's organized, with Mr. Dawson suggestion that magicians would likely be separated into those who serve the Crown, those who work for an organization, and those who operate independently.