What is the proper string depth in a nut?

I thought with all the expertise here, this would be a good place to ask this question.

I have been experimenting with strings lately, and with the differing string diameters it got me to wondering where the string should actually sit in the nut? I have seen online where it was said that ideally it is half in the nut and half out, is this basically correct?

Unfortunately, the fluorocarbon strings I have been trying are thinner than what came strung on the ukes and now sit all the way in the nut.

Is it preferable to just fill a nut and have it re-filed or replace the nut completely once I settle on the strings I like?

This article and website might help. Looks like there isn't a definitive depth as long as it's functional.

String height off the frets is important to playability and to intonation. Depth in the nut is a lot less important. If they are not deep enough they can roll out of the slots. If I had a uke and was happy with the height of the strings above the frets I would not worry about the depth into the nut, unless it was too little and they rolled out.

That said the first time I made a nut I got it way too high. By the time I got the string height right they were so low in the nut that they looked stupid so I lowered the top of the nut.