Are businesses and individuals buying clean / green / renewable energy contribute to slowing down the climate change?

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.

Two things:

1. The price reduction on the polluting good helps consumers of the polluting good, but also hurts producers of the polluting good. This results in less output and, in the extreme, market exit by the least-efficient polluting firms. A full analysis would be needed to show which effect wins in the end as far as overall emissions go.
2. This is accelerated by imposing a Pigouvian tax on the polluter, both by raising the price of production for polluting firms and through revenue-recycling (which, when done properly, either raises incomes, accelerates the fall in price of green energy options, or some mix of both).

Of course, this is all airy-fairy stuff. A Pigouvian tax is challenging to enact, because on the global scale, enacting one is a kind of Prisoner's Dilemma. Hence output-based allocation systems for high-emitting, trade-exposed industries right or wrongly advocated on the grounds of "competitiveness".

Higher demand for green energy =less demand for other energy it is true , but demand for energy in developing countries is so high that any energy whether it is green or not , it's not their fault or unawareness it is their necessity , green energy costs more as the cost of green infrastructure ( maintenance , capital infrastructure ) is huge . Eco tax , regenerative industry models , secondhand good industries, carbon credits for industries could make a difference in climate change . German eco tax initiative in 2003, California's carbon cap- trade scheme in 2013 are proofs that green energy contributes to climate change