No, 'software engineer' is not exempt. The article says "Engineer is a protected title in certain contexts in B.C.", and links to the association page, which has this:
Software and computer engineering have been designated as disciplines of professional engineering since 1999 and 1998, respectively, and are currently listed as disciplines of professional engineering under the Regulation.
Non-registrants are prohibited from using titles like “Software Engineer” or other high-technology-related engineering titles such as “Firmware Engineer”, “Hardware Engineer”, “Data Engineer” because they imply that the individual using them is a registrant and that they are authorized to practice the professions of software or computer engineering in BC.
We do, however, recognize that not all software development constitutes software engineering. In some cases, individuals mistakenly use a title like “Software Engineer” when they are not actually engaged in any engineering work. Individuals may have roles more appropriately characterized as a “developer”, “designer”, or “programmer". For more information on software engineering and for help determining whether work constitutes software engineering requiring registration with Engineers and Geoscientists BC, please see our software engineering applicant webpage.
Funnily enough, both provincial and federal government doesn't mind contracts and applications with title "software engineer" for immigration related services. As long as work experience fulfils the NOC 2173 definition. Licensing in the NOC is only mentioned as required for Professional Engineer work.
That private organization is the one that licenses "engineers" in BC.
The entire Canadian regulatory system is crazy anachronistic with a bunch of legacy colonial and British cruft. It's like the worst parts of American federalism were mixed with the worst parts of British centralization.
Engineering societies in Canada are not really private. They are considered self-regulating professions, where the society has an official role regulating their profession. Each of the provinces have enabling legislation that authorizes these societies to create binding regulation within the bounds of the legislation.
Edit: added missing word