Not an ATT user - Buying a locked ATT iPhone 13 Pro and paying in full, can I immediately unlock?

You cannot unlock it immediately. I’m not sure where you saw that information on here. There are requirements that need to be met. 

https://www.att.com/support/article/wireless/KM1262649/ it says it here on the 2nd line, but has lots of contradicting infomation afterwards.

Terrible idea!  You'll wind up paying at least $100 more for the phone. 

If you purchase a new device that is locked to AT&T there is a minimum 14-day requirement. It will not be unlocked until it is past the return date.

If It is a brand new account there will be a 60- day wait.   Not only does the phone have to be fully paid for but the account must be in good standing. If the account is in default AT&T will not unlock.

If you're adding a line to an existing account just to bypass the 60-day wait., there still has to be paid service, activation charges, and you will pay for a full month's service on the line.

In general AT&T stores will not let you purchase outright. They will require that you add the line and activate the phone as an installment. You won't be able to pay off the installment until it shows up on the bill.  If you received the phone and don't activate it, it will not register and therefore will not be in the unlock portal and will be declined. 

There are people who ordered the iPhone 13 Pro during the pre-order and still don't have the phone in their hands so I would find it very unusual and unlikely that you will have better luck ordering from AT&T, do not trust their website, you might find a physical store that has them in stock.

    I would think the best place to purchase one, without extra payments, and without having to wait up to 60 days, would be Apple. 

Yeah, I see that line.  That page has a last edit date in October, meaning they may have just changed it.  I read that line the same as you.  However, I know the policy did not previously contain the line

Was paid in full when you bought it (not on an installment plan). You won’t have to wait 60 days to unlock it.

@JefferMC

So are they essentially dropping the new account wait term?  

That cuts the time in half, because you're still paying for a month's worth of service and at least 14 days to get past the return date.

Sure looks that way.