Further confirmation of 8 April 1763 is provided by a contemporay account in The Letters of Horace Walpole, Earl of Orford, Vol. 3, "Letter 150 To George Montagu, Esq." (Friday, 8 April 1763), pp.208-9:
Lord Bute resigned this morning.  His intention was not even suspected till Wednesday, nor at all known a very few days before.

Wednesday would have been the 6th, and that accounts for the few sources that end Bute's term that day.  "This morning" refers to the 8th.

10 Downing Street explains "The Missing Days" between Bute's resignation on the 8th and Grenville's succession on the 16th.
There can often be breaks in the holding of the office of the Prime Minister, usually if the handover of power follows an election defeat or the death, resignation or retirement of the current PM.

All in all, at this point, I'm fairly convinced that 8 April 1763 was Bute's last day as PM.