It's the nut. I recently bought one of the Jimmy Page Telecasters and it did the same thing. I took it to my guitar tech and he fixed it in in a couple of minutes by reshaping the slot in the nut very slightly.
It could totally be the nut if the sound occurs when hitting the open string, but if it happens when fretted, I'd discount that as the problem and look at other things, depending on where on the string it's happening: I'd check the position of the truss rod first, then I'd switch saddles, at least on the Strat, which would involve less time if you have a traditional Tele bridge, then I'd check for very very subtle high frets.
The big question is, where does the problem occurred on the high string on each guitar? Does it happen for either on the open string? When fretting a particular note? Does it happen when fretting a range of notes?
I.e., in the first instance I experienced this, with a high fret on my Les Paul, it was essentially fret buzz that was so unbelievably subtle that it didn't sound like fret buzz at all; it was a terrible pinging sound that felt like my guitar hated me. I ended up spot leveling that fret and a couple others, which corrected the problem. I also removed the onboard buffer much later, just because I weirdly missed my Paul's inherent muddiness, which also yielded this warmth that was totally lost with too much clarity haha. I'm curious if you roll back the tone knob on your Strat and your Tele if you'll hear the problem go away. Before looking at adjusting your tone, I'd see if both guitars might have really subtle setup issues. The reason I say that is, you can totally take away an ice pick sound with the tone knobs on the amp, your guitar, or whatever else would tame the brightness, but then of course you can't use that wonderful tone you've created! So if you do love a tone that's bringing out ice, I think you can probably keep that tone and correct a potential problem with the guitars at the same time, allowing you to continue using bright tones across the board, no pun intended. If you have access to a good tech, that might be a good place to start too, if you don't normally do your setups. It could possibly give you that wider palette to work with, minus the pain! Best of luck!
On the other thread, the way I read it, it seemed almost like it's everything you do on that string that sounds thin. If that's the case, maybe posting a sample would help more.