How can I prevent this from happening every time it rains in my yard?

If the retention pond was installed by the local government, they should be responsible.

If from neighbor, you'll need to increase height of your property where it adjoins their's, to make it equally high, or higher than your neighbor's.

Start by adding a large amount of composted manure, perlite, and top soil to the site, digging at least 18" deep. Create a drain via swale or piping to divert all this water. You cannot divert it to a neighbor, but it could be done via dry creek or French drain system, or Hydro Blox system, instructions for which are found on You Tube. Hydro Blox can handle much more water faster than a French drain.

The condition of the area would indicate it hasn't been touched in years, so soil could be bad, or simply unable to drain due to the vegetation.

If it were me I would think about a french drain. Here is a link to one you can do yourself.

https://www.bowerpowerblog.com/easy-diy-french-drain/

Most places it's not legal for your neighbors to run drain water onto your property . This could be a major conflict to work out w/ your neighbors , cause arguments ! Whoever put in wall should repair it , hard to do if it's an old wall . Looks like it's near tree roots ? These could be messing up both drain system and wall .

I did buy the exact french drain you bought, then I brought it back to the store. After reading your story and seeing the results, I'm going to Home Depot again to try this.