Just to add, we are talking about the "Inter-Cities Fairs Cup" here, which, as it's name implies was a cup organised around cities that had trade fairs. Unlike the Champions' Cup and Cup Winners' Cup which, from the start, you qualified for by being national champions or cup winners, there was no qualification as such for the Fairs Cup and entry was to cities by invitation only.
In it's early years, composite teams representing cities entered (a London XI were runners up in 1958) as well as club teams, but the latter "represented" their cities. Clearly, the Fairs' Cup committe, in 1962, were still operating on this basis and wanted representatives of 3 big cities from England - Liverpool, Sheffield and Birmingham; as such 1962 runners up (Small town) Burnley had no chance. On the other hand, by nominating 2nd, 4th and 5th, the League were clearly indicating that there should be some merit basis to entry. However, as the Fairs' Cup committe ran the competition, it was up to them up got to enter.
Given this, United did not "qualify" for the Fairs Cup in 1962 as no-one qualified. You were invited to enter. However, the fact that Wednesday rather than United were initially invited to represent Sheffield, despite United finishing higher in the League does beg a few questions. I can only assume that Wednesday being "bigger" (can of worms etc - but much higher attendances and league runners up the previous season) weighed heavily with the committee.