None of the three is what shows up in your account. What lands is the housing assistance payment, which is the authority's share of the rent, and the tenant pays you the rest directly.
The relationship, with round numbers. Fair market rent is a published estimate for an area and bedroom size, used as the basis for setting policy. Call it $1,300. The housing authority then sets its payment standard somewhere in a permitted band around that figure, which is how you get $1,430. The payment standard is the most the authority will use in calculating the subsidy.
Gross rent is your contract rent plus the published utility allowance for whatever utilities the tenant pays directly. If the allowance is $130 and your contract rent is $1,300, gross rent is $1,430. Gross rent is the number tested against the payment standard, so if the tenant pays their own utilities, your collectible rent is below the payment standard by the amount of that allowance.
Then the split. The tenant's share is calculated from their income, commonly around 30 percent of adjusted monthly income, and the authority pays the difference between that share and the gross rent, less the utility allowance. So on a $1,300 contract rent you might see $1,000 from the authority and $300 from the tenant.
On the naming: the program is the Housing Choice Voucher program, and "Section 8" is the older name people still use for it. Loosely, "Section 8" also gets used for project-based subsidy, where the assistance is attached to the building rather than to the tenant. Those are different arrangements with different contracts, so when someone offers you a "Section 8 building," ask which one they mean.
The piece to add to your model is that the tenant's share is a real collection line. It's your job to collect it, and it isn't guaranteed.