Interesting... It's a few years since I was operational and I vaguely recall monitor being introduced into standard phraseology. At that time the clear distinction was that Contact meant change frequency and check in on the new frequency whereas Monitor meant change frequency and keep quiet until you're called.
I don't remember the idea that it should be used to listen to the ATIS or whatever - maybe it was introduced later or maybe it just didn't go into my head (because it doesn't seem very intuitive). Either way, having now checked some more of the books, the UK text reflects the current ICAO procedures so we are not talking about a UK anomaly.
In the books the meaning of Monitor is 'Listen out on (frequency)'. The guidance 'An aircraft may be instructed to 'Standby' on a frequency when it is intended that the ATS unit will initiate communications soon and to 'Monitor' a frequency when information is being broadcast thereon' seems to create the ambiguity that you have raised.
Given this background I think it's a matter of opinion as to who's right. But like you, I can't see the point of instructing an aircraft to to listen out on box 2 so I tend toward my original interpretation....but equally, in the absence of any more detail, I could accept someone interpreting the instruction differently.
I guess it beautifully illustrates the benefits of standardisation and the problems caused by poorly written rules!