Ummm...my interpretation is correct. Peter had said “even if all deny you, I will die with you”, and when the test came his weakness was revealed. So now he is ashamed and hesirmtant to make such a claim again. Remember Jesus said “ live-streams thou more than these. That is why Peter uses the weaker word phileo instead of agape. But the threefold denial was followed by the threefold question and the threefold commission, feed my sheep. And a prophecy that Peter would stand in the future, even unto death
This Jesus telling Peter that he meant what he said when He said earlier, “when you are converted, strengthen the brethren”, now stated as “feed my sheep”