I would suggest that the most important point is that the term "Humidifier" is inclusive of all types of humidity producing devices and this can be misleading given that not all such devices are the same.
The main thing that separates each device in terms of effectiveness (ability to help your fathers breathing) is the amount of water vapor they are able to produce.
Keep in mind that a mist is not the same as water vapor.
A mist is actually a dense cloud of tiny water droplets, each of which are very tiny but still large enough to see and large enough to transport pathogens from the water chamber to the human airway - Hence the need to change the water in a cool mist "Humidifier" and reduce the transportable pathogens.
Water vapor is simply the gas form of water, one oxygen atom and two hydrogen atoms flying around independently , obviously it can not be seen, cannot transport pathogens, it is in the air all around you all the time and the human airway is specially equipped to add water vapor to the gas we breath before it gets to the lungs, this is a natural conditioning process that occurs every time you inhale.
A true and effective, water vapor humidifier will only ever aim to produce water vapor, not mist or steam, to supplement this natural gas conditioning. It can be very useful in preventing the effects of airway dehydration that you will see being caused by dry oxygen use (water vapor humidificayion can be used simultaneously with oxygen delivery via some nasal prong systems) and it can be very usefull in providing active hydration of the airway to mobilise secretions.
Water vapor can not be produced without water and heat energy, so all such water vapor humidification devices are called heated water vapor pass over humidification devices, sometimes also called active humidification devices.
The most effective of these devices will be able to deliver fully humidified gas at a dew point temperature of 37 degrees Celsius (core body temperature). This means that there is no further water vapor for the airway to add to the inhaled gas and therefore no dehydration can occur in the airway which could thicken secretions.
A study published in a medical journal called Respiratory Medicine and authored by Harry Rea and colleagues in April 2010 found that using water vapor humidification in the home for one hour a day on top of any other medication etc. resulted in fewer and less severe exacerbations in patients with COPD. It also suggests that the higher gas flows which are tolerable given the humidity levels may help to encourage slower and deeper breathing which would have also contributed to the result.
I am not aware of any studies which have shown a benefit to COPD patients from cool mist or non heated humidification.
In summary, if it was my father I would not hesitate in encouraging him to use a humidifier provided it was capable of producing high levels of water vapor humidity (aka 37 degree C dew point or there abouts, at least more than 33 degree C dew point anyway)
I would also encourage him to use it as often as possible at high flow rates (it's just humidified air remember) and in conjunction with any other medication his doctor has prescribed.
I hope this was of some help.