Ability to manually save state

**Is your feature request related to a problem? Please describe.**

If I know a particular action changes something significant enough to warrant saving the state to disk, I want to be able to request a save (similar to how you can use `ctx.request_repaint()` and related functions if you know a repaint will be needed).

**Describe the solution you'd like**

The idea that comes to mind is adding a `ctx.request_save()` or similar, which triggers the same behavior as if the autosave duration was reached.

**Describe alternatives you've considered**

I tried storing the creation context passed to the `eframe::App`, but that caused ownership problems.

I tried storing a reference to the `eframe::Storage` instead, and also ran into ownership problems.

~~I can't even figure out a messy way to do it with existing functionality.~~ (**EDIT:** I found a way to implement it. See https://github.com/emilk/egui/issues/5243#issuecomment-2406186520.)

Start by tracing the eframe::App creation context and eframe::Storage usage, using the existing ctx.request_repaint() behavior as the API pattern. Determine how the autosave-duration path is triggered and define how a manual request reaches it without ownership problems. Done means an application can request a save explicitly and it follows the same behavior as an autosave.

Written by the indexing model from the issue text.