Ddriver, good work on asking when you are not sure.
This subject is sometimes poorly understood even by people flying airliners.
A common mistake is to think that the speeds used to determine the 'EDTO area of operations' is the speed that you should fly at. This is not the case at all. In the 737's that I fly the OEI speed that is used to calculate the maximum diversion distance is M0.79/310kts and that gives an area of operations of 825nm for 120min EDTO. Quite a few pilots I fly with think that they should fly these speeds if they lose an engine in the cruise. That is not the case. There is nothing to stop you flying those speeds if they are what you deem appropriate in the circumstances as the EDTO diversion fuel is calculated using those speeds, but it is up to you and your Captain or First Officer ( ultimately the Captain) to decide on what speeds are most appropriate at the time.
Is it most appropriate to descend at Vmd? If you are trying to maintain altitude due terrain or traffic then probably yes.
Is it most appropriate to descend at M0.79/310kts? If you need to get on the ground ASAP due fire or critically injured pax etc probably yes, but take into account the new altitude you will have to be at. Will it put you in icing or a front?
Is it most appropriate to descend at an intermediate speed and transition into LRC? If you want a margin ( to stall and over speed)due to turbulence and to arrive overhead with more fuel in the tanks then probably yes.
You can see from the above scenarios that each situation is different and requires thought. It is a good idea to think prior to the situation developing. For example, if you are making an ocean crossing there is a good chance you can rule out descending at Vmd and just use your current Mach number into the IAS for LRC if no traffic is about.
we hold the altitude till we get to the Vmd
That is nice in theory but I would have a think about what your priorities are before deciding that is the highest priority. For example, there have been incidents ( and I can think of one accident) where trying to achieve this has resulted in loss of control. Keeping your aircraft flying safely is your number one priority and if you are very high, beginning a descent may be your highest priority and you may only have 4 or 5 seconds to initiate it before being 'behind the drag curve' and things starting to unravel.
Know what your actions will be when you see an engine roll back.
Know roughly ( within ten knots) what your OEI LRC speed will be and exactly what your Vmd is.
Be careful to do your own research rather than relying solely on what one Captain or Check Captain tells you as the advice given is not always of the standard it could be. Ask several trusted people.
I think your question is a very good first post.
Framer