So, what, exactly, is a tropical period?
Is the US Naval Observatory wrong somehow here? Have mistakes been made? Or are there misquotes?


That's a matter of opinion. In Duncan Steel’s opinion, yes, they made a mistake. In the US Naval Observatory's opinion (and also that of Her Majesty's Nautical Almanac Office), there is no mistake. The Nautical Almanac defines a tropical year as the time from one mean vernal equinox to the next. Steel is using a more pedantic definition. Increased pedantry does not always equate with increased correctness.


  Is it possible to explain in everyday English the geometrical difference between a "Vernal Equinox year" and a tropical year or a mean tropical year?


Right now, Northern Hemisphere spring and summer (Southern Hemisphere autumn and winter) last longer than do Northern Hemisphere autumn and winter (Southern Hemisphere spring and summer). This situation will be reversed in about 13000 years. These changes are due to the Earth's axial precession. That the Earth undergoes axial precession has multiple ramifications, three of which are that:


The length of a tropical year (no matter how one measures it) is less than is the length of a sidereal year.
The times from one March equinox to the next, one June solstice to the next, one September equinox to the next, and one December solstice to the next are not constant.
The times from one March equinox to the next, one June solstice to the next, one September equinox to the next, and one December solstice to the next are not equal to one another.


Elaborating on the third point, the calculated length of one mean December solstice to the next was 365.242740 days (ephemeris days) on 1 Jan 2000 while one mean June solstice to the next was 365.241626 days, or about 1.6 minutes shorter than the duration from one mean December solstice to the next.

So what is a "tropical year"? The US Naval Observatory, which has a much longer history than does Duncan Steel, defines the tropical year as the passage from one March equinox to the next. The US Naval Observatory inherited this definition from Her Majesty's Nautical Almanac Office, which in turn inherited it from the Church of England, which in turn inherited it from the Catholic Church. One of the key drivers of the Gregorian Calendar was getting the date of Easter correct. (Getting the date of Easter correct was very important to many Renaissance minds.)

Duncan Steel defines the tropical year otherwise, as an average value of the times it takes for the Sun's longitude as perceived from the Earth to advance by 360°. That's fine, but to argue that people who use a different definition is not just wrong. It's "wronger than wrong". The fact is that there are multiple definitions of what constitutes a "tropical year", all of which are valid.