As I understand it, the Saturday of Easter Week, in the ancient Gelasian and Gregorian Sacramentaries, is the 'Close of Easter'. The Gregorian collect of that day talks about us having celebrated the Paschal Feasts (paschalia festa egimus), and Gelasianum numbers the following Sundays as 'after the close of Easter'.
The post-Conciliar reforms made much of Easter being 50 days long and being one single Great Day of Festival. They renamed the Sundays as 'of Easter' rather than 'after Easter, and chucked out the old collects for the Sundays after Easter (their best hope for any sort of survival was to be assigned to the season per Annum) because they didn't consider them 'Paschal' enough. To replace them, they cobbled together a set of collects which was substantially new. They gave their game away by transferring the Collect for the Sunday after Easter (with its talk about now having finished the festa Paschalia) to the Saturday before Pentecost.
The Church of England, with its Liturgical Commission dominated by 'Bubbles' Stancliff and passionate as ever for any passing popish bandwagon, drove the tendency even further. The addition of Alleluias to Dismissals (which even Bugnini's collaborators had confined to the Octave of Easter) was extended to the whole Fifty Days. A number of variations in the liturgy, to mark and enhance the unitary nature of the Fifty Days, was confected and embodied in the C of E's new "Common Worship".
I wonder just how securely founded in both the Bible and the patristic traditions, of West as well as East, this newly minted view of Eastertide is. It certainly seems to be true that the reforms of the 1970s represented a new divergence between the customs of West and of East: by levelling out Eastertide we lost the ecumenical convention, which we shared with Orthodoxy, of marking the unique character of this one very special week by allowing it to retain a whole lot of unique (mostly archaic) liturgical features. The Byzantines delightfully call it 'Bright Week' (I resist the temptation to repeat all the information in the Wikipedia entry sub hac voce) and they make the service each day to be completely unlike that of any other week of the year. One example in our Western idiom of thus making Easter week 'strange' was the traditional Western disuse of Office Hymns during this week; in place of them and of other elements in the Office, we used simply to sing the anthem Haec dies. Considering the enthusiasm with which the 'reformers' orientalised so much of the Roman Rite, it seems extraordinary that in other respects, such as this one, their concern was to drag the West out of a usage common to both of the Church's 'lungs'. But then, they always did what suited their own immediate whimsies.
There is an even profounder 'ecumenical' aspect to this question. S Paul assumes the familiarity of his largely Gentile Corinthian congregation with the Jewish usages of a seven-day Passover Festival celebration in unleavened bread (Exodus 12; Deuteronomy 16; I Corinthians 5). This suggests that the Paschalia festa, that is, of Easter Sunday until Easter Saturday, represent not only Apostolic practice but are part of the immemorial continuities linking the Old Israel with the New. Which would make the post-Conciliar alterations seem even more irresponsibly capricious and 'anti-ecumenical'.
One final point. As in Judaism and in Byzantine usage, so in the pre-Conciliar West, this very special week ended on the Saturday. We then gave up the Alleluias in dismissals, and the proper Communicantes and Hanc igitur. But in the Novus Ordo we are supposed to continue them on the Sunday, Low Sunday, before saying farewell to them.
As I understand it, since the Saturday in Easter Week was the Clausum Paschae, the Sunday after it, the English Low Sunday, was the First Sunday After the Close of Easter. So when Traditionalist Catholics and Prayer Book Anglicans call the following Sundays 'After Easter' they do not quite mean 'After Easter Sunday', but, technically and pedantically, 'After the Great Seven-day Festa Paschalia which stretch from the Easter Vigil until they "close" before the First Evensong of Low Sunday'.
I am not, of course, suggesting that the remaining six weeks before Pentecost should completely lose an 'Eastertide' status. As Dix puts it, "After the Pascha the 'great 50 days' ... were already recognised [at the end of the second century] as a continuous festival, during which all penitential observances such as fasting and kneeling at corporate prayer were forbidden, as they were on ordinary Sundays also. ... just as for the Jews the fifty days of harvest between Passover and Pentecost symbolised the joyful fact of their possession of the Promised Land, so these fifty days symbolised for the Christian the fact that 'in Christ' he had already entered into the Kingdom of God. Like the weekly Sunday with which this period was associated both in thought and in the manner of its observance, the 'fifty days' manifested the 'world to come'."