The requirements for electrical permits are outlined in the Electrical Safety Regulation, under Part 2 — Permits, Inspections and Regulated Products / Division 1 — Permits. The Electrical Safety Regulation can be found on the Government of BC website at:
https://www.bclaws.gov.bc.ca/civix/document/id/complete/statreg/12_100_2004#part2
Section 11 clarifies that a permit is required when electrical work is performed on electrical equipment. An electric hot water tank would be considered electrical equipment and would fall under Section 11.
11 (1) When electrical work is performed on electrical equipment, a permit is required
(2) A permit is required for any electrical work that requires the attendance of a utility to connect or reconnect an electrical service.
The only regulated products that may be replaced by a licensed contractor without a permit are covered under Section 18, and an electric hot water tank is not one of the items listed.
18 (1) No permit is required if the only electrical work performed is testing.
(2) An individual may replace the following regulated products without a permit, up to a maximum rating of 150 volts to ground, with electrical equipment of a similar type or rating: