I paid 5.88 or 6.03?
I think both are correct, you have taken screenshots at different times.
The fee you pay is in BNB, so you cannot convert the net transaction fee in $ because it always changes according to the BNB price.
(Your total spend BNB * 0,25%) + your BNB for gas fee = Your net fee
(0,0091841238583506 BNB * 0,25%) + 0,000797922007368 BNB= 0,0008209 BNB
Calculation in rate 1BNB= $589.01
Dear, both screenshot are taken at just 1 minute difference. But I still agree with you that during this time, BNB price changes and this there can be difference in USD price.
BUT, if you check both transaction then FEES IN BNB are different in both screenshot. in one it shows 0.000797 and in one it shows as 0.001015