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At first I couldn't the difference between the two ways you were pronouncing it. But now I realise that the understated 't' way of pronouncing it is just like my cousins' accent - they come from Plymouth, England. They never pronounce the 't' in words, most notably water, which in their accent sounds like war-er or wah-er.
In my normal accent I'd say Peyton with the t because that is how it is said where I am from, however when I am around my cousins - and I do pick up their accent immediately when in their presence - I wouldn't say the t but that's just because of the accent.

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