How to install window screen without screen track

It does look like that a screen could fit where you're pointing.

I've included a picture of some screen hangers and a lock. The hangers would go on the top of the screen and trim allowing the screen to be lifted into your trim track. The locking device would be screwed into the center of the sill and mounted to hold the screen securely into the trim. Any water that accumulated during rain would be able to run under the screen. The hardware would be permanently attacked to the upper frame, screen and sill so it could be caulked and sealed to prevent any water from migrating into the wood.

Typical approach for modern screens/windows (as Jack has shown the very vintage approach) is that the screen has a frame (typically aluminum) that has springs on one side or two sides so it can be fitted into the slot by compressing the springs, and when released is held in the slot by spring pressure.

The sill is "trackless" so it won't trap water and create a problem. The screen frame being held in by the side grooves seals it well enough if it's properly fitted.