You should not need to call two different agencies to close your VFR flight plan. Call the nearest ATC unit, FSS (or a FIC) or a CARS is the appropriate action except IFR flights to an aerodrome served by a control tower or FSS where you are not required to file an arrival report unless requested to do so by the appropriate air traffic control unit (CAR 602.77 (3)).
Section 3.12 of the Transport Canada Aeronautical Information Manual (AIM) echoes the regulations from CAR 602.77 (1) and states:
In order to comply with CAR 602.77, an arrival report for a flight plan shall be submitted to an ATC unit, an FSS (or a FIC) or a CARS as soon as practicable after landing but not later than:
(a) the SAR time specified in the flight plan; or
(b) where no SAR time is specified in the flight plan, one hour after the last reported ETA.
This idea of reporting to the closest unit is also supported by ICAO Annex 2 (Rules of the Air), standard 184.108.40.206:
When no air traffic services unit exists at the arrival aerodrome, the arrival report, when required, shall be made as soon as practicable after landing and by the quickest means available to the nearest air traffic services unit.
Perhaps what is missing from the AIM/CARs is the idea that an arrival report should be filed with the closest ATC unit, FSS/FIC, or CARS. This is because the country receiving the flight (Canada in this case) becomes responsible for alerting services at destination (since it's located within a Canadian Flight Information Region). This concept aligns with the standards in ICAO Annex 11, standard 2.10.1:
Flight information centres shall be established to provide flight information service and alerting service within flight information regions, unless the responsibility of providing such services within a flight information region is assigned to an air traffic control unit having adequate facilities for the discharge of such responsibility
I can't explain why 1800WXBrief would have seen the flight plan "closed" after a few minutes but responsibility for it would have indeed been delegated to Canada after departure. This might appear as closed to them in their system.
The call your FBO received from NAV CANADA was likely from FSS or the nearest flight planning office at the Area Control Centre asking whether you had arrived since they had an open flight plan for alerting services. Once the FBO notified them you were safe, they likely closed the flight plan.