Is it true that if you take cold showers that you could potentially have these benefits?

It's not likely to have a quantifiable or definitive scientific proof. If you like taking cold showers then take them. Those almost all sound like overblown marketing or advertising.

Cold followed by heat or vice versa does stimulate circulation for certain. When you damage tissue through an injury docs recommend an ice pack and heat or vice versa to help get the circulation to increase delivering the healing power of oxygen. However for normal every day...cold shower wakes you up and moves the blood around. That characteristic is proven medically.

I always take a hot shower and, something I learned from a past life, finish with about a minute of as cold water as I can stand it. It does referesh you and makes one alert. Doesn't help spelling though

Your blood might move around when cold; mine coagulates.

Heat then cold makes the blood circulate a bit more...especially that morning shower.