I answer a lot of questions on Quora. Many of those questions are about automotive things that may be of interest to CarNewsCafe readers. Like this one.
My answer: There are three basic types of battery terminal used in most automotive. Both the Society of Automotive Engineers (SAE) and Japanese (JIS) posts are the conical “up off the top” type you’re likely referring to. The other is the side terminals with screw-in receivers.
In both the SAE and JIS standards, the positive is the larger post and the negative is smaller. This is to prevent accidental cross-attachments (reversals) wherein the negative is accidentally attached to the positive and the positive to the negative. That can result in a seriously fried electrical system on most vehicles.
That, in a nutshell, is the real reason for them being different sizes.
Note that JIS termianls are already smaller than the SAE terminals as well, so if you purchase a JIS-style battery to fit in a vehicle designed for SAE, the terminal ends will not tighten down onto the posts. Similarly, JIS-designed ends will not fit at all over SAE-sized posts.