I am a public adjuster and we see this quite often. Condo water damage claims are one of the most complicated claims out there. There are a lot of moving parts here so please reach out to me direct if you need more info.
First, you need to review the HOA/COA CCRs and see who is responsible for damage to property by insurance perils. Many times the HOA/COA will cover walls, framing, exterior etc. however we also see in a lot of properties that the HOA/COA is actually liable for all property that was damaged with the exception of contents and any improvements and betterments you or a prior owner made to the unit. We look at the CCRs, state statutes and insurance policies to see who is actually responsible for the damage. If the HOA/COA is responsible for insuring all your property, file a claim with their insurance company. Be prepared for them not wanting to do this. It also depends on the amount of damage and their deductible. Even if the damages do not exceed their deductible, the HOA/COA may still be responsible for the repairs. This could provide coverage for both affected units.
However, many policies do not cover back up of sewers or drains and this coverage needs to be added to a policy.
It sounds like your insurance is going to cover your claim and that would be great for you. Since you were not negligent for this damage, (sounds like a back up and due to no fault of yours or your tenants) your insurance would not be legally liable for the damage to the unit below. if your insurance pays for your damages, they may actually go after the HOA/COA policy to recover what they paid out (subrogate) if they feel the HOA/COA should have covered it. More common, we see the unit owner insurance deny coverage and tell the unit owner to pursue against the HOA/COA.
Hope this helps.