I am afraid I put my scientists hat on in response to a question like yours
@SueCuz and ask questions rather than provide answers!
The big question is, what do you (you or your consultant or your DN) want to get from your measurements?
So what are you going to get from your consultant's suggestion? Your blood glucose goes up and down through the day and by testing when you get up and before meals what you will get is some indication of your minimum blood glucose because in theory, it is at its lowest at those times. That might be useful for diagnostic purposes or for adjustment of medication but will tell you little or nothing about the highs.
What are you going to get from your DN's suggestion? By testing at random times during the day you will get some idea of how much your blood glucose varies, the highs and the lows. Bit better than your consultants suggestion when it comes to characterising what is going on in your system but again most useful for diagnostic purposes and medication adjustment. Only comment I would make is that it would take a long time at 2 tests every other day to get enough information to make any sensible decisions unless your blood glucose was very high.
If you want to use the measurements to try and control your diabetes then you need to be testing before and after meals to see how the things you eat change your blood glucose levels. You can then find the things that give big changes and then change your diet to stop it happening.
My suggestion is to try and satisfy all three! Aim for three or four tests a day.
On a three test day, test on waking and then test immediately before a meal and two hours after. Rotate around your meals. On a four test day, check before and after two meals again making sure that over a few days you will roughly test the same number of breakfasts, lunches and evening meals. Keep going for a couple of weeks and you will begin to get the picture.
That way you can give the consultant what they want, the DN what they want, and if you keep a food diary, you can find out the the things which give you big rises in blood glucose which is something you can use to control things. Wins all round, except maybe for the DN who will be worrying about the number of test strips being used and their cost!