Why does the white siding not cover that exterior wall?

My friend's solution to her problem which is the same as yours:

A French Drain.

https://www.houselogic.com/organize-maintain/home-maintenance-tips/french-drains-when-you-need-them/

Call the basement waterproofing guys please.

Why does the white siding not cover that exterior wall?

Because there is brick/stone there.

Get it...looks like exposed wood clapboard on my monitor. Thought it could be part of the problem.

When we were looking for our first house, all of the houses we saw in this one area (early to mid-20th century builds) either had French drains or water damage in the basement. Maybe it's the way they built basements then, or a high water table, or the combination of the two. But based on what you're saying, I recommend you look into it. I don't think it will be inexpensive, but it beats dealing with wet carpet all the time, or even just four times a year.

Were your previous neighbors okay with you rerouting your water drainage to their property? Our city ordinances allow us to slope our property for proper drainage, but we can't let that drainage become our neighbor's problem.

We have a high water table. The great but expensive solution was drain tile.

Here's the thing about water: it seeks level. So in the ground, that you can't see, water will travel horizontally under your plastic sheeting. You got lucky at your last house (and had an agreeable neighbor as @CeeWhy points out). It's just a matter of how much water is infiltrating the ground, how much already exists (water table), and how quickly it can drain to somewhere lower (within the soil strata). If you are experiencing a climate now like most areas, it's not that rains may come more often or less often, but that either case is more extreme when they happen. So when it rains, there is probably more times you have to move a lot of water right away and your plastic sheet cover (or concrete slab, or any impervious surface) doesn't account for what can happen below it.

Your basement won't stop leaking unless you dig outside and redo the entire foundation with waterproofing measures, which is unrealistic for most. Attacking it from inside with those so called "basement experts" has so many negatives, those methods should only be a very last resort. You can start with outside measures like addressing existing grade slopes, downspouts, swales, and french drains. If you only have extreme rains 3-4 times a year, those methods may be enough. Each property is different though, so much more information would be required to advise on exactly how those methods would be conceived.