At the end of the Holy Communion service in the Church of England Book of Common Prayer are 9 rubrics (i.e. rules or notes) the seventh of which begins:
And note, that every Parishioner shall communicate at the least three times in the year, of which Easter to be one.
Prior to the Reformation, constitution 21 of the Fourth Lateran Council, 1215, required communion to be received at least once a year, at Easter. This applied to the whole Roman Catholic church, of which the Church of England was then part. Receiving Communion meant eating the bread. It did not refer to simply hearing mass, attending the service, which was expected much more frequently.
Although the Prayer Book rule requires that people communicate three times a year it only specifies Easter as one of them, the other two are not stated.
You slightly misquote from your reference when you refer to "triannual celebration of the Eucharist" - it in fact says triannual reception. It is not a reference to the number of times Holy Communion was celebrated (i.e. occurred) but to the number of times per year each person was required to receive (eat and drink) the bread and wine. In Cathedrals and colleges Communion was held at least weekly but, particularly in Georgian times, only a few times a year in rural parishes.
A fairly common practice in the Church of England was for people to attend and receive communion at Christmas and Whitsun as the other two occasions apart from Easter, and generally in Georgian times these festivals had communion services. The Synod of Dort, in Holland in 1619, in Rule 63 said Holy Communion services should be held every two months and also at Christmas, Easter and Pentecost (a traditional name for Whitsun). This was not in any way binding in England, but illustrates a popular understanding of the three. The Church of Scotland, by contrast, celebrated Communion usually once or twice a year, but did not associate it with Christmas, Easter, Whitsun, or any other festival.
In the early 20th century most Church of England churches had communion weekly but these services were not, except on special occasions, well attended in comparison to Matins and Evensong. The three times a year rule could not be enforced and its observance would be by a specific strata on the scale of commitment between those who received more frequently and those who received only at Easter or (the majority of the population) not every year at all. The character Durham, or rather his mother, fell into this strata, at least as regarding minimum expectations of her son. An additional consideration is that Christmas is at the end of the year. A person who got to December having communicated only twice in the year, and who wished to respect the rule, or whose mother wished him to respect it, would have to communicate by the end of the year to make up the three. This is an additional reason for Christmas Communion, which may have applied to the character Durham.
Receiving at Christmas and Whitsun, especially Christmas, was merely the customary way of fulfilling the three times a year. Only Easter was, and is, mandated.
Since that time communion has become the main (best attended) service in most Church of England churches, and in many is the only regular service. This change has coincided with a fall in the number of regular churchgoers, but the extent, if any, to which this may be cause and effect is unclear.