Upvote:6
The Roundheads had various names but in general did recognise it as a civil war, more precisely, "civil warre" or "seven yeares’ warre".
On Civil War, in a letter from General George Monck to Major-General John Lambert:
I desire your Lordshippe to consider seriously the sad consequences of a civill warre, least you involve your self too farre in itt, that wee that have drawne our swordes together in defence of our countrie’s peace and safety may nott sheath them in one another’s bowells.
On Seven-Year War, by the English Puritan clergyman, Thomas Edwards, in Gangraena, 1946, p.352 - (I know you mentioned during period of Protectorate):
Both the power of the Kinge and Lords,” argued another, “was ever a branch of tyranny, and if ever a people shall free themselves from tyranny certainly itt is after seven yeares’ warre and fighting for their liberty.
Finally, as for your concern that these were "names that would be favored by the royalist side", the following has interpretations of the Civil War that you might find useful: