(1) Why was the name "Extraordinary Form" introduced by the Holy Father in the motu proprio Summorum Pontificum? Answer: in order to solve a liturgical law conumdrum.
Traditionally, throughout the history of the Church - at least since the differentiation of rites became clear and attached to specific patriarchies and geographical areas - bi-ritual priests have been exceptional. They still are an exception. Additionally, the Pope felt the need to finally undo the injustice that had been kept - and defended by most canonists - since the advent of the Apostolic Constitution Missale Romanum, of Paul VI (1969), that had created the Novus Ordo Missae: had it, and the previous and subsequent documents that modified all rites of sacraments, abrogated the Traditional Roman Rite?
The use of the term "form" solved both problems: it did not make all priests in the Latin Church, including the vast majority of secular priests, immediately bi-ritual (in law), which would be rather untraditional; and, most importantly, it solved the apparent problem of the impossibility of the abrogation of a liturgical rite of immemorial origin. (It was an apparent problem because, as the Pope implied when he said that "what earlier generations held as sacred, remains sacred and great for us too, and it cannot be all of a sudden entirely forbidden or even considered harmful," the immemorial liturgical Rites and Uses of the Latin Church could not and cannot simply be abrogated.) In a sense it is an artifice, a noble intellectual construction, since the common celebration of the Traditional Latin Mass and the Novus Ordo Mass seem to express two very distinct rites - but the use of such legal constructions is quite common in law, and there is nothing immoral in it. The use of the terminology made clear that celebrating the Traditional Mass is a solemn right of each priest of the Latin Church.
(2) Despite this, the expression "Extraordinary Form" is NOT the "official" name of the Traditional Roman Rite. It is just one of the many ways to refer to it. In fact, as can be seen in the very texts of the official documents, several different names are used to refer to the Traditional Roman Rite.
The motu proprio itself speaks in its first words of the "extraordinary use" and of the "ancient form" (antiqua forma) of the Roman Rite. In its articles, mention is made of "the Roman Missal promulgated by St. Pius V and reissued by Bl. John XXIII" (that is, Missal of St. Pius V also is as "official" as "Missal of Bl. John XXIII" - no wonder Cardinal Navarette-Cortes used the term in 2008); it is an "extraordinary expression" (extraordinaria expressio), and also "extraordinary form" (forma extraordinaria). It is also called by the motu proprio the "earlier liturgical tradition".
The rites of sacraments according to the Traditional Rituale Romanum are characterized as according to the more ancient ritual (Rituale antiquior), same adjective applied to the Pontifical, and to the form itself: earlier form (forma antiquior).
All these names are included in the short text of the motu proprio Summorum Pontificum itself!
In the letter to bishops, mention is also made of "the Roman liturgy prior to the reform of 1970". The Pope says in the letter that they are not "two rites" (though in the letter he uses the name "new rite"! - making us quite comfortable with also using the expression old rite...), but also uses different names for it therein: a "usage", the "earlier Form", the "1962 missal", the "old Missal", the "ancient Latin liturgical tradition" (a very beautiful name, by the way)
In the Instruction Universae Ecclesiae, preference is given to the expression "forma extraordinaria", but also there all kinds of different expressions: "usage", "use", "Usus antiquior", "1962 Missal"...
These are just the "official" names used widely in the documents themselves - not forgetting the need for clarity that demands a continued use of expressions that are established in the vernacular, such as Traditional Latin Mass (TLM) in English, and "Tridentine Mass" (even if not particularly exact) in English and in several European languages. Not to mention the very respectable use (for instance, by former President of the Pontifical Commission Ecclesia Dei Cardinal Castrillon Hoyos) of the expressions "Gregorian Rite" and "Classical Roman Liturgy".
THEREFORE: (1) do not feel forced in any way to use the name Extraordinary Form as if it were the only acceptable name - it is not even the exclusive name used in the documents themselves;
(2) do not complain when others use it, as if it were illegitimate or unacceptable; if you do not like it, fine, just do not use it yourself.