Pipe size is 1/2" - 13 mm - but the fitting is 15 mm or 9/16"
The normal shutoff in the States is 3/8" and is not compression fittings unless you want them to be. The hardware stores sell these neat little connectors that adapt from the machine to the shut off. So if it uses a 3/8" tube, it will have a 1/2" or 9/16" nut or perhaps even larger to make it possible to hand tighten.
I am sure that there is a name for this type of connector, but it is not a 'compression fitting although it uses compression by the outer cap to conform the bulbous tip onto the shutoff which has a shape to accept the tube end. A compression fitting generally has a ring of material that is collapsed or compressed onto another material to effect the seal between the two and the ring then also seals to the output of the shut off. The material of the ring is usually copper or plastic. I doubt that you have one of those. They are usually more expensive and less common.
I looked at the description that you posted and you appear to be correct. These are definitely not what we normally run into. It appears that a compression fitting is being used to connect between the line coming in and the feed to the dish washer - in fact if I read that correctly, it looks like an ordinary garden hose connector for the dish washer.
But are you sure that you have to cut the line? The only way to T in is to cut the line to install this connector - is that what you are proposing needs to be done? Seems a bit like overkill. Usually there is some kind of fitting where a threaded T could be installed so that two shutoffs one for whatever the run was for and one for this unit. Otherwise, I would suggest that you install a T of the normal variety such as a copper sweat fitting if that is what is there, and not use the compression valve, just install a normal shutoff onto a regular T like your system uses. (If plastic use plastic, if copper use copper, etc., install the plumbed T and then install the shutoff.) I would avoid the compression fittings - but that is just me.