Are businesses and individuals buying clean / green / renewable energy contribute to slowing down the climate change? I'm not sure if an individual choosing a "green energy supplier" is making any difference.

My (potentially naive, uneducated) logic is as follows: 

higher demand for green energy = less demand for other energy

Less demand, therefore lower price, those who are polluting the most now have fewer incentives to switch. Supply and demand, elementary market forces.

I would like to better understand if choosing green energy makes any difference?



Personal note: I'm in the camp of effective altruism. In my version of reality, giving £1 to a homeless is a selfish "feel good" act that does solve the underlying problem of education, employability, affordable housing, mental health support, resilience and life skills.
Your intuition is correct, actually switching from dirty good to clean good will also increase demand for clean good so price there might increase as well. However, this does not mean doing some voluntary change could not help as it also depends on how the quantity demanded changes with price (elasticity of demand). So switching is not necessarily unhelpful. 

However, at the same time it is also only a partial solution. In order to fully  internalize the externality it’s necessary to apply Piguvian tax to the dirty good at a level that fully internalizes the externality to people consuming the good.