The "Mark" designation had its historic root in equipment for ships: in the early 20th Century for the artillery (sic) of US naval ships, by World War II for their radar instruments.
Considering that Jefferies (and others) were war veterans, there were no "Mark" designations for ships in the 1960's and the context in which the primary phaser schematic is presented (to tell the viewer what kind of phaser he is looking at, not what kind of ship), I'd say it's pretty clear that the "MK IX/01" designation refers to the primary (long range) phaser.
I'm sorry but it isn't as cut and dried as you are making it seem. The MK designator is used for all sorts of military gear, naval guns etc. But Matt Jefferies, a military pilot in the European Theater during WW II, would have been familiar with the British use of the term for aircraft. The MK designator was used by them for British military aircraft such as Typhoons, Hurricanes, Hellcats, Spitfires, Lancasters etc. Spitfires were produced in a MK IX generation. In fact, IIRC they were further broken down as MK IX/01, 02 etc.
I think Jefferies is responsible for assigning the term to that phaser illustration. But it is just as likely that he is referring to a starship as it is he is referring to the phaser type. There is simply no way to tell beyond inference from the fact that as a pilot it is more likely he was referring to the starship than some naval gun.
When I first read the Matt Jefferies interview about the Enterprise being "the first bird", "the first in the series" my conditioning immediately wanted to place the Constitution before the Enterprise and then my inner voice took over: What the heck are you doing, isn't "first bird" clear beyond doubt?!?. YMMV, of course
You joined this BBS just last year so you might not be aware of the fact that I was arguing the "Enterprise-class" argument ten
years ago. Just out of sheer orneriness.