You've got three separate documents and they have three different sets of parties.
The voucher is issued by the public housing authority to the tenant. It's the tenant's document. It says this household has been found eligible and can look for a unit, and it lists things like the household size and how long they have to find a place. You aren't a party to it. You might see a copy so you know it's real.
The lease is between you and the tenant, exactly like any other rental. You write it, you both sign it, the housing authority isn't a party.
The HAP contract, housing assistance payments contract, is between you and the housing authority. That's the document that says the authority will pay you a stated amount each month on behalf of that tenant in that unit. The tenant doesn't sign it. So the person who told you it's the lease had it backwards, the lease and the HAP contract cover different relationships.
The tenancy addendum is the fourth piece and it attaches to your lease. It's federally required language that becomes part of the lease you and the tenant sign, covering things like the grounds for termination and the fact that the tenant only owes their calculated share of the rent.
The part that catches people out is timing. The HAP contract generally isn't executed until the unit has passed inspection, so you can have a signed lease and no payments yet. Ask your housing authority in writing how they handle the lease start date relative to the inspection pass date, because their practice sets when your money actually starts.
