I am addressing the cited claim (not the different form stated at the top of the question)
Do not mix raw and boiled water! Grandma was right. Just why mixing of boiled and raw water leads to a rapid emptying of the bowel, the science cannot explain yet. But the fact is obvious.
The World Health Organisation (WHO) recommend that water be raised to a temperature of 100° C for 60 seconds in order to kill off bacteria and viruses.
When you pour water at 100° C into a cold container such as a cup (or teapot) at 20° C, the water reduces in temperature to about 80° C by the time you've finished pouring (I've measured this but have no reference at the moment)
Studies in the UK have found that the average temperature for drinking tea is between 56° and 60° C
Clearly, if you add cold unboiled water to the cup of 80° C water in order to reduce the mixtures temperature to 60° C, that new water is not going to be raised to 100° C for 60 seconds. Therefore any bacteria it contains (if any) may not be killed and may lead to an illness such as the diarrhoea suggested in the question.
Science has no problem explaining this commonsense advice for people with unsafe water supplies. The difficulty in comprehension only arises when the claim is over simplified and separated from its context.
It is not the mixing of cold and hot water that creates a health threat out of nothing. It is the failure to follow a sterilization procedure that would treat an existing health threat.
Some travel advisors say
Russia Travel Health Tip: Don't Drink the Tap Water. Even locals do not drink the tap water in Russia. Tap water may be even more dangerous for foreigners. Not only will it contain elements that a traveler's body may be unused to, it also may also cause diseases due to being improperly sanitized."
This suggests that the safety of drinking water may be a subject of concern to Russian citizens to a greater extent than in countries where laws successfully require tap water to be safe for drinking.
mixing boiled and unboiled water can be dangerous for one's health, even if it is safe to drink both liquids apart.
The question contains no citation for any claim with that proviso.
To support this claim you'd need something like a study that takes two containers of water, one boiled one unboiled, and tests three groups of people (with significant numbers in each)
- a group that drinks only a cup of the boiled water
- a group that drinks only a cup of the unboiled water
- a group that drinks only a cup of mixed water (say 50/50)
To support the claim, the third group would have to show a statistically significantly higher incidence of health problems than either of the other groups.