I'm looking to replace the traditional Chorus headset on my old Pinarello as the bike is about to get a Spring-clean.
I'd like a Record-style hidden one, but no-one seems to make them for 1", only 1 1/8th.
anyone know better please?

Don't think that can be done and as far as I know 'hidden' headsets arrived long after 1,1/8" came about. There isn't enough beef in the headtube or clearance there to get a race in on standard 1" frames.

1" Record headsets look pretty classy I reckon.

I seem to remember an exception when 1" was around - it was a Cinelli(?) road or track frame that had a unique hidden style headset but it never caught on.

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I looked myself in the last days for a 1" headset (new steel frame hand made by a little frame builder in Italy, not really WW) and didn´t see any hidden one (... or they were hidden too well... )
In the end I decided for a Ritchey WCS.

Please someone correct me if I'm wring on this one, as I am not 100% sure, but I'm pretty sure that a hidden semi-integrated style headset requires a special head tube. For it to be hidden, the bearings actually need to fit into the entire headtube, which on a classic 1" headtube steel frame simply cannot work. If you just place the steerer into the headtube, and look at the clearance around the sides, theres no way you can fit bearings and a bearing cup in between there.

Out of all my bikes, only one has a hidden headset, but the headtube is flared out on the top and bottom, actually making it impossible to install a traditional headset.

Correct.

The type of headset that will fit is determined by the frame when they build it. So no, you can't convert from a conventional 1" to hiddenset.

Much as you can't convert from a conventional 1 1/8th to hiddenset either.

(there are a couple of places that make convertors, but these are usually to make a headset smaller, i.e. lefty/headshock to 1 1/8th convertors and the like)