Your own source gives a pretty complete answer:
1630s, Scottish borrowing of French cachet "seal affixed to a letter or document" (16c.), from Old French dialectal cacher "to press, crowd," from Latin coactare "constrain" (see cache). Meaning evolving through "(letter under) personal stamp (of the king)" to "prestige." Compare French lettre de cachet "letter under seal of the king."
This indicates that there was an Old French dialectical meaning of cacher as "to press", which evolved into the Modern French word cachet, meaning "a seal affixed to a letter", which was then borrowed in 16th century Scottish with the same meaning.
The entry on cache:
1797, "hiding place," from French Canadian trappers' slang, "hiding place for stores" (1660s), a back-formation from French cacher "to hide, conceal" (13c., Old French cachier), from Vulgar Latin *coacticare "store up, collect, compress," frequentative of Latin coactare "constrain," from coactus, past participle of cogere "to collect" (see cogent). Sense extended by 1830s to "anything stored in a hiding place."
Shows the English cache originating from a French-Canadian back-formation from the French verb cacher to the noun cache, meaning "a hiding place for stores".