> I don't know much about the state of rechargeable battery technology in the late-1800's
This link has some info about rechargeable batteries in the context of electric vehicles. Basically in 1881 key improvements to the original 1859 lead-acid design made batteries practical on board a vehicle at an industrial scale.
> Credit to that generally goes to Carl Benz, who got his patent in 1886.
The above link reiterates the date of 1884 for a production vehicle by Thomas Parker, two years before the Benz patent.
Also, earlier than that, a tricycle was successfully demoed along a Paris street in April 1881, following a 1867 two-wheeled prototype that did work but wasn't quite drivable for production use.
EVs got a slight head start to ICEs but they subsequently developed alongside each other from the late 19th to early 20th.
>> Credit to that .. Carl Benz, .. patent in 1886.
> date of 1884 for a production .. two years before the Benz patent.
The car was produced in 1885, the patent was granted in 1886.
The way I understood the OP's "The original cars were electric..." comment was that these electric cars (plural!) were dominant for some time until they were replaced quite some time later ("over 20 years before the Ford Model T") by cars with internal combustion engines.
As far as I know, that didn't happen. Electric cars and cars with internal combustion engines happened essentially simultaneously, with steam a century earlier (1769). From the Wikipedia article the OP quoted (section Golden age):
"In the United States by the turn of the century, 40 percent of automobiles were powered by steam, 38 percent by electricity, and 22 percent by gasoline. "
So even during the "golden age", >60% of cars were not electric. I have to admit I was surprised that steam-powered cars were that widespread.
There were earlier self-propelled vehicles.