Actually you're both right. The last 3 banks I've had have had the $500 limit because if your bank doesn't 'natively' support zelle, Zelle themselves limit it to $500 per week taken via your debit card. If your bank integrates zelle, then the limit seems to be up to the bank up to the $5000 or whatever you quoted.
This can be a major problem, because some landlords only take their rent by Zelle and if you accidently zelle someone at the wrong time and your whole household runs out of Zelle limits you're fucked for paying the rent.