Simplified path. The puzzle showed the main steps; the full route is being checked.
Latin
malum persicum
‘Persian fruit’
into Late Latin
Late Latin
persica
‘The peach’
into Anglo-French
Anglo-French
pesche
‘The fruit’
into English
English
peach
‘The familiar fruit’
The story of peach
Peach preserves part of the Latin expression for a Persian fruit. Late Latin persica passed into French forms such as pesche, then into English. The name records an association with Persia, not a botanical claim about where the species originated.
These are selected stages, not every spelling or intermediary. The story identifies omitted links; older forms did not all disappear at once.