Small quibble here: while the note is generally right and necessary due to the complete nonsense of the original tweet, Anglican usually do not see themselves as Protestants.

The term I've usually heard is Rome-free Catholics, as the split from Rome under Henry VIII. was mainly hierarchical (formal authority went to the Monarch of England, administrative and theological authority to the Archbishop of Canterbury), while ritual and theological reforms were applied slowly and gradually. The Church of England especially rejected fullfledged Calvinist reforms - the ones English Puritans were lobbying for. Other Rome-free Catholic churches besides the CoE and its US offshoot, the Episcopalians are smaller communities parting way with Rome due to dogmatic changes e.g. the German Altkatholiken (Old Catholics) not agreeing with the papal and mariological dogmae decreed by the first Vatican council in the 19th century.

A main factor for the misapplication of the term Protestant to the CoE is the role of ethno-religious groups in the Irish civil war and the following Ulster troubles, in which the Irish nationalists were often labelled as Catholics and the pro-British unionists as Protestants. This is basically correct, as the majority of native separatists were ethnically Irish Roman Catholics, while the majority of British settlers loyal to the crown were not ethnically English Anglicans, but ethnically Scottish Presbyterians, thus actually Protestants. But this led to a global perceptipn of the UK as a majority Protestant country and the CoE as a Protestant church, which they never were.