I am Jamie.
If you heat a paperclip with a lighter and push it into the floor it will melt a vinyl but not a linoleum as linoleum/marmoleum are composed of natural materials.
My guess is that its vinyl from the pictures but hard to tell. One thing with linoleum is you never get a brilliant white colour which the marbling looks to be in your picture which would suggest vinyl. The practical test will tell.
Hire a scrubber drier for it. It will make the job a lot easier. Even if its just to go over after a scrub with your rotary to pick up and neutralise the floor.
If you were to go with a nylon pad on your machine you could get a diamond pad like a twister or equivalent that would buff up the floor a little when you run the scrubber dryer over. This would let you use a higher grade pad but bring the colour up as it takes out the marring from the scrub pad.
If you are going to apply a surface coating to it go for Chemspec Tight Shine or Pioneer Eclipse Floor Matte as they are both suitable for sports environment and very forgiving when it comes to application method.
Happy to discuss if you want to give me a call on 01334656787 I have done hundreds of these over the years.