The single engine cruise speed used for ETOPS calculations varies from airline to airline, as each airline can choose the cruise speed they will use for their calculations. Once the cruise speed is chosen, it is used to determine the area of operations (the circles on the ground in which the operator is approved to fly) - and those circles are
fixed. That is, they don't change with daily changes in wind or temperature. The trade-off is in choosing a faster single engine cruise (and hence a wider area of operations) the fuel flow increases significantly (and the amount of fuel carried to cover the worse case diversion - which
is planned for each flight, taking wind and temp. into account.)
As such, the speed must be one that the aircraft
is capable of, and the fuel carried must be enough to support a diversion at that speed. These are
purely planning considerations. On the day, the PIC may choose any scenario required to safely return the aircraft to the ground.
What is important is the intention to fly the airlines ETOPS speed. What you actually fly during the diversion is up to the PinC on the day. You can expect to be questioned by the authority why you deviated from your airline published ETOPS approval.
By briefing anything different your instructor is correct that you are not meeting the ETOPS certification requirements held by your airline. You can however brief different failure case options but the standard ETOPS speed must be the primary option you intend to use, i.e. that is the approval your airline operates under.
This is simply incorrect. the only time you can "expect to be questioned" is if you choose a scenario which uses more fuel than you have available at the time - which has nothing to do with ETOPs, as you can expect the same questions any time you run low on fuel.