The user tests is wrongly assuming what our API is doing - please find below my comments on the user code with fixes followed by simplified example:

[ContextMenu("PrefabTest")]
private void PrefabTest()
{
// Create a serialized representation of "this" object
SerializedObject so = new SerializedObject(this);

// Find the serialized representation of TestString property
var property = so.FindProperty(nameof(TestString));

// Set the serialized representation of TestString to Time.time.ToString()
property.stringValue = Time.time.ToString();

// Apply the serialized representation of TestString to "this" object
// NOTE:
// This doesn't record the override in the Prefab instance and HasPrefabInstanceAnyOverrides will return false!
// To record the property override in the Prefab instance one needs to call PrefabUtility.RecordPrefabInstancePropertyModifications
so.ApplyModifiedProperties();

// Record the modifications in the Prefab instance
PrefabUtility.RecordPrefabInstancePropertyModifications(this);
Debug.Assert(PrefabUtility.HasPrefabInstanceAnyOverrides(PrefabUtility.GetOutermostPrefabInstanceRoot(this), false));

// Apply Prefab instance modification to the Prefab asset
PrefabUtility.ApplyPrefabInstance(PrefabUtility.GetOutermostPrefabInstanceRoot(this), InteractionMode.AutomatedAction);
Debug.Assert(!PrefabUtility.HasPrefabInstanceAnyOverrides(PrefabUtility.GetOutermostPrefabInstanceRoot(this), false));
}

Simplified version:

Please find below a simplified working example:

[ContextMenu("PrefabTest")]
private void PrefabTest()
{
var prefabInstanceRoot = PrefabUtility.GetOutermostPrefabInstanceRoot(this);

TestString = Time.time.ToString();
PrefabUtility.RecordPrefabInstancePropertyModifications(this);
Debug.Assert(PrefabUtility.HasPrefabInstanceAnyOverrides(prefabInstanceRoot, false));

PrefabUtility.ApplyPrefabInstance(prefabInstanceRoot, InteractionMode.AutomatedAction);
Debug.Assert(!PrefabUtility.HasPrefabInstanceAnyOverrides(prefabInstanceRoot, false));
}