I agree with you in the most part. In most situations I would think that an indoor pool within a strata building would NOT technically be classified as a “public pool”.
If the pool is located on common property then the occupants may use it in accordance with any by-laws conditions. All the lot owners jointly own any swimming pool or spa pool that is on common property.
Members of the public are not entitled to access the common property of your strata scheme unless invited by an occupant. Invitees must comply with all conditions imposed by the by laws and the Act.
Although strata pools are not technically “public pools” they would still need to comply with the various Acts that govern pools e.g. the Swimming Pools Act 1992 for safety issues.
The quote that you provided is from the Public Health Act 2010 . Environmental Health Officers do not regulate public health issues in pools and spas within private residential premises including strata residential buildings. However, an OC has a duty of care to provide a safe and healthy environment for users of the common property pool.
See here for more information on pools in strata: