You have misread the articles (or read misleading articles).
Epigenetic modifications are, by definition, any modification to the DNA and proteins attach to DNA that does not affect the nucleotidic sequence (note that I here ignore some of the difficulties defining the term epigenetics). Such modification include methylation of histon tails for example (a histon is a protein around which the DNA is wrapped). Those epigenetic modification can appears in reaction to the environment and can be passed on to the offspring giving it a seemingly Lamarckian style of inheritence.
A mutation is, by definition, a change in the sequence of nucleotides. An epigenetic modification is therefore not a mutation (and a mutation is not an epigenetic change).
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Some theorists believe that random mutations are more rare and time consuming than a mutation that develops as a response to change of the environment the organism is in.
You might want to avoid calling "mutations" "random mutations" because it feels like you might not understand how unclear this phrasing is. Have a look at the post Are mutations random?
Epigenetics change are NOT mutations.
The claim is too general and unclear and is impossible to comment about it without knowing what process you are interested in. One might claim that most genetic variance in populations is caused by epigenetic variation or make other claim that can be discussed but the above claim is too broad and unclear.
So does this mean a couple living in say, Siberia (both of whom migrated from say, Congo) will have higher probability of transferring gene variations to their children or grand children that will allow them to adapt better to the cold.
I don't understand what you mean here. And it feels to me that explaining everything that is unclear and going through all possible ways of understanding this sentence would take too much time!
Does that explain how humans were able to evolve quickly to their travels from Africa to the rest of the world and still survived the drastic changes in environment, rather than conclude that random gene variations led to adaptation?
You are really not using the term random correctly, which makes your question hard to understanding. Also, instead of gene variation I suppose you might want to use the term allele.
It is possible that epigenetics has played a role to allowing faster range expansion / range shift of human populations. I don't know if this is the case and I don't how important of an effect it would have.
Is this genetic adaptation as a response to the environment biologically proven?
Epigenetic, not genetic. Yes, epigenetic modifications do exist. We have plenty of evidences. I don't know how much we know about their impact on population range shift and survival but I would notice that it depends upon what mechanism exactly we want to consider into the broad term of epigenetics. For example, everybody will consider histon modification as part of epigenetic modifications. However, not everybody will want to consider fat content in the egg as an epigenetic modification. There is also a blurry border between what would be environmental effect (such as maternal effect) and epigenetic effect. But this is a discussion for another time.