If it was a standalone booking on BA metal bought as a Basic fare then you can buy luggage on top but it doesn't generate the Silver allowance by simply buying the first bag allowance. To do that you need to but either a Plus (if shorthaul) or a Economy Standard fare (longhaul). So it's just the one bag, and I'd very surprised if anything online would give you a different impression, the BA website spells this out very clearly in my view. The other flights on the trip are irrelevant if this a standalone booking, since to use BA's jargon, they are separate contracts. And the reason it is just 1 bag is that you are still on an HBO fare, just with the segregated extra bag. You can buy more bags, at ever rising cost. To get the full Silver allowance you need to buy a ticket which has a checked baggage allowance.

The 32kgs issue is a bit unclear since it can be read in different ways. Personally I think you would probably get it if it was a main BA station on mainstream BA metal.

Usually if you have checked luggage it doesn't make sense to go HBO on the fare, it is priced deliberately to make it infeasible. The exceptions relate to things like AA and AY metal.