Looks more like the wild plums here in Texas, the leaf also looks thinner like a plum. Most persimmons I have seen when I was a kid, the fruit looked less round and slightly "bell" shaped. Either way may be a nice little "goldmine" for a pie!!
Definitely not like our persimmon trees or fruit. Our leaves are short and broad and shaped similar to a sand dollar. The fruit is still very green and hard and won't be ripe until late fall or even early winter. If you happen to find a fruit bearing plant DON'T eat one that's not yet ripe as they are astringent and will suck every bit of moisture from your mouth in seconds (don't ask me how I know).