The high altitude explosion Wikipedia link is worth looking at.
When I say that I doubt the 1961 Soviet missile intercept happened, here is what i think happened: The Reds may have launched an anti-missile missile that passed within a rather wide radius of a descending target missile's terminal stage, and the inerceptor's atomic bomb may have exploded in the ionosphere upon radio command from its ground radar tracking station.
However, that so-called "intercept" was an unimpressive technological event, even back in 1961.
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So here was the plan, as I interpret it from some PeePruners: Suppose the Next Big War got started in mitttel Europa, and things weren't going well for the good guys. On the thrid day of this, British decision makers would be wanting to let go with tactical nukes, but the USA might be reluctant to do so.
Then UK Prime Minister would call up the White House and say,
"Terribly sorry to have to bring this up, but if you Americans won't join with us striking the Reds on the Continent in Germany or - er, are they in France now? with our smaller tactical nuclear weapons .... if you won't agree ...
... We'll launch our Polaris missiles now! "
Was that the strategem?