Can pilots tell if a cloud or fog is made out of water droplets or ice crystals when flying through it? I understand that clouds can be made out of water droplets or ice crystals. And that fog is nothing else than low-lying clouds.

However, I've experienced fog in a wide ranges of temperatures. I like mountaineering and you often get fog high in the mountains. I've been lucky enough to visit the high mountains of Antarctica and I've also experienced fog there at very low temperatures.

This answer (in the comments) and the fog wikipedia article mention that ice fog happens only at extremely low temperatures (-35 or -40 C). But I think I've experienced fog at those temperatures too. Some times, with the right light I've seen the air filled with tiny crystals, but I wouldn't call that "fog", because it is very thin.

Then, as far as I can tell, all thick fogs/clouds feel and look the same. No matter if they are made out of ice crystals or water droplets. Is that right? Or is there a way to differentiate between ice/water fogs?

Or maybe contrary to what I think, I've never experienced frozen fog.

But, would I notice the difference if I were to ever experience it? How would they feel/look different?

Other sources, like the UK meteorological office say that water always remain in liquid state in fog, no matter how low the temperature is. Is that true? Is it true for clouds too? In that case there would be no such thing as "ice clouds".

One possibility is that ice or water clouds would give rise to different optical phenomena, like sun dogs that are produced by ice crystals, but usually in conditions were they are not dense enough to be considered fog, like in this image:



(Image from Wikipedia page for sun dogs)

So I wonder what's the experience of pilots, as they go through clouds?. When they do so, they know also the external air temperature, which is a plus. Do they see sun dogs or other optical phenomena that differentiate water from ice clouds?
Boeing provides some decent notes on this in their powerpoint which you can find in full here. Their sections on pilot reports and the summary section have some good notes for identifying ice crystals in flight/clouds/fog 


  
  Pilots have reported rain, sometimes ‘heavy rain’ on the windscreen, Sometimes at impossibly cold temperatures. This is
  believed to be the result of these small ice crystals melting on
  impact with the heated windscreen
  No observations of significant airframe icing
  Turning on landing lights at night and seeing reflective precipitation .It is likely they were seeing the larger particles even
  though they make up a smaller fraction of the population
  Hearing a different sound from rain. Ice crystals hitting the airframe sounded different than rain or hail
  Light to moderate turbulence
  No significant airframe icing
  TAT probe frozen
  Ice detector not detecting ice (when installed)
  Appearance of rain on the windshield
  


The same powerpoint also discusses that satellite imagery can be used to see ice crystals. A lot of planes even small stuff have the ability to receive feeds of this data which can be used to identify sections of ice crystals in flight. 

According to various sources the threat is particularly dangerous due to the fact that there is no hard and vast way to visually identify the crystals and in many cases the effects are similar to other events and may lead the pilot to belive something else is happening.

For pilots we can only go by look, unless you are flying slow in a 172 you cant exactly pop the window open at altitude and feel the cloud. Simply put pilots can not reasonably tell if a cloud is vapor or crystal prior to flying into it and observing the above noted actions.