Your pergola will almost certainly be attached to Common Property at some point and this may require a special resolution by-law which will cover such items as ongoing maintenance of the pergola (your responsibility) as well as ensuring that a future Owners Corp doesn’t change its mind and ask you to remove the structure. You’d do well to just accept that and get a strata lawyer to draft the required by-law for approval. The assurances you are giving are basically what a by-law would provide but it would also protect your and future owners’ of your property’s interests.
The issue of council approval depends on the council but they usually take little interest in it – especially if it is approved by the Owners Corp and doesn’t affect other buildings.
As far as changing her advice, the Strata Manager is clearly not sure of the laws herself and in any case, can only act as instructed by the OC. The final advice sounds right and I would proceed along the route of getting a special resolution approval, subject to the OC approving the design and materials (if need be) and you employing a reputable builder.
Meanwhile get bthe council’s planning department to tell you exactly what they need from you and do your best to comply.
It’s tricky but it’s better to be street legal than fighting either your neighbours or the council.