There are 2 types of Persimmons, The Hachiya which you describe and they MUST BE SOFT ripe to eat and then they are pure ambrosia. Then there is the Fuyu which you eat like and has the crunch of a ripe apple. The Fuyu has a flattened appearance and the skin san be tough.
From another source ... "An entirely different species, Diospyros kaki, the Japanese persimmon, originated in China, leaped to Nippon thousands of years ago, then, in the early 1800s, came to California, likely with Asian immigrants. Two varieties fill store shelves, both resembling medium-sized orange tomatoes. Hachiya (which dominates the market) is pointed at the bottom and tastes somewhat astringent. Fuyu has a flat bottom and a sweeter taste. Neither has the rich flavor of the smaller native American fruit."
Your info got me to looking up persimmons and I think I'm learning something here .... thanks! :-)