It's the second one. On a 1,200 unit you might get 850 from the housing authority and 350 from the tenant, and those numbers move depending on the household's income.
How the split gets set: the authority calculates what the household can contribute, generally around 30 percent of their adjusted income, and pays the difference between that and the contract rent. That means the split isn't fixed for the life of the tenancy. If the household's income drops, their share drops and the authority's share rises. If their income rises, the reverse. Some households have very low income and their share is small or zero, which is why you see people describe the whole rent as government paid.
Why that's still steadier than a market rental. The larger piece of the rent arrives from an institution on a schedule, and it keeps arriving when the tenant loses a job, which is exactly the moment a market-rate tenant stops paying. The piece you're chasing is 350 rather than 1,200, so a bad month costs you a third of what it would otherwise.
The thing that trips up new landlords: you cannot charge the tenant more than their calculated share, even if the market would bear more than the contract rent. Side payments are a program violation. And on collecting the tenant share, you generally can pursue nonpayment through your normal state eviction process, but the tenancy addendum limits your grounds for termination and your state's process controls the steps, so read the addendum and check your local rules before you rely on that.