Anyhow, I made a random search in the text of MBh and found more than thousand times the name Karna is used in different Parva-s and Chapters and only the name VasusheNa is used only in the context of narrating reminiscence of the episode for certain only to mention his earlier name. The occurence too in the total text is limited to only 11 occurences of which there may be some other persons also known by that name.
It seems that this was mentioned only to show the derivation of the word Karna deliberately or वैकर्तन (whether included later or at the time of the original composition) and while the whole Mahabharata was retold, he was already popular by the name KarNa. That much can be certainlly said.
The same verse, where in one version cited, it seems to define वैकर्तन as noticed by Nilakantha, who explains वैकर्तन otherwise, is found in Geetha Press Edition as follows:
ददौ वासवाय
तथॊत्कृत्य प्रददौ कुण्डले ते; कर्णात् तस्मात् कर्मणा तेन कर्णः||
Though this justifies the definition कर्ण, it seems he had only one ear with two rings.(?). Somehow it seems totally a cumbersome verse trying to say something else, but not we cannot get.
The verse quoted in BORI edition, appears with certain variations in Geeta Press;
पुरा नाम तु तस्यासीद्वसु षेण इति श्रुतम् .
ततो वैकर्तनः कर्णः कर्मणा तेन सोऽभवत् .. 21..\\
as follows:
प्राङ्नाम नाम तस्य कथितं वसुषेण इति क्षितौ। .
कर्णो वैकर्तनश्चैव कर्मणा तेन सोऽभवत् .. 31..\\
The above verses appear even with the reading वैकर्तनश्चैव in the Geetha Press Edition and translated he was known thereafter by this act, by both names वैकर्तन and कर्ण which agrees with the definition as given by Neelakantha.
Out of curiosity, I have encloded a word document with all the 1751 occurences in Mahabharata, many of them are clearly refer to him even during the conversations in the earlier Chapters than the Vanaparva episode. In the Aadipravarva, it is appended with and
MBh, 1, 104, 11.2
सहजं कवचं बिभ्रत् कुण्डलोद्द्योतिताननः /
अजायत सुतः कर्णः सर्वलोकेषु विश्रुतः / ||
and here the time of the episode is not mentioned, but he was suggested by Surya beforehand, in his dream to exchange when he is asked for Kavaca Kundala, with the Shakti, even if he do not reject his request. This is mentioned as "some time" in his life:
ततः काले तु कस्मिंश्चित् स्वप्नान्ते कर्णम् अब्रवीत् / (17.6)
आदित्यो ब्राह्मणो भूत्वा शृणु वीर वचो मम / (17.7)
and immediately after that he was requested to give his sole protection and he gave them in exchange for Shakti. (which was used against Ghatotkaca and Arjuna was saved from it).
It is much more plaussible, and etymologically feasible too he got because of his ear-rings in his ears and in which case above statement seems to be redundant. In niew of this, the idea suggested by Srimallalitalalita seems to settle the matter that by this act कर्ण became वैकर्तन though both are derivable with reference to the episode.
विकर्तनस्य = सूर्यस्यायं वैकर्तनः। = the one born of Sun विकर्तन;
or
वि+कृन्तति - इति कृती छेदने - विकर्तनः - (the cutter of his own ears and body), विशेषेण कर्तनाद् वैकर्तनः (as given by Nilakantha in his commentary).
विकर्तन एव वैकर्तनः।
कर्णौ - प्रशस्तौ जन्मना कुण्डलाभ्यां युक्तौ अस्य इति कर्णः। can be the derivation also from the word कर्ण - meaning ears.
किरति = विक्षिपति इति कर्णः - also can be the etymology derived from the verb कॄ विक्षेपे.
The name is used 75 times in the text of Vaikartana (in the Digital Corpus ofSanskrit) including the ones in the cited verses also. whereas the name कर्ण is available in 1751 occurences while the name वसुषेण occurs only 11 times and the name आधिरथि is used 35 times, राधेयः - 134 times. The statistics is from the data-base Digital Corpus of Sanskrit.
Here is a general assessment of the episode of giving the ear-rings and armour.
Another account for his in-built armour and ear-ring is to be found in the narration of nara and Narayana, episode in BhagavataPurana(I could not locate in the original text). According to it, he was the incarnation of a demon Sahasrakavaca, who had been protected with 1000 kavaca-s which were to be pierced only after fighting 1000 years. In turn Nara and Narayana, managed to cut off 999 armours and then he fled and took refuge in the Sun God and o was reborn as Karna with his remaining armour. With the armour he is impossible to be defeated with any weapon and hence he was deprived of his kavaca, in the guise of Brahmana, by Indra to facilitate his defeat this birth also at the hands of Arjuna (Nara's rebirth).
More about this concept of incarnation of Sahasrakavaca, to exmphasise the protection of his kavaca and kundala, there are two blogs:
But there is no such reference in Mahabharata itself, I think. Though Narayaneeya, a southern composition, mentions of this Sahasrakavaca, killed by Nara and Narayana, it doesn't mention anything of his re-birth as KarNa. But it is popular in the southern version of the story as pointed out in the above article.