When an event listener( notifier ) works with multiple classes and you want to limit it to one class you have to supply the eventClass.
app.notifiers.add("slct", new File("~/desktop/selectLayerHandler.jsx"),"Lyr ");// edit this path to your event script
The notifier should run when the layer is selected in the layers panel before it is moved. Deselecting the layer then selecting it again should run the notifier. It does for me on my system.
Or are you talking about a selection on a layer? Because clicking on the document should not deselect the layer in the layers panel.