Does White Tea have less caffeine, or better yet, does White Tea have no caffeine? I get this question all the time and the simple answer is no. At least, not for the reasons you're thinking and not in the way you're thinking. Let me break it down.
First off, there are two main compounds within tea when discussing caffeine levels. The first is of course caffeine. The second is a compound unique to tea called L-Theanine. This second compound is important to understand as it binds to your receptors prior to caffeine and slowly releases it into your bloodstream. This is one of the reasons for why people rarely get "the jitters" with tea, as opposed to coffee.
The other important thing to note is that caffeine is minimally affected by temperature in the processing of the tea. L-Theanine on the other hand IS impacted. Very basically, more heat means less L-Theanine, but not necessarily less caffeine.
One last thing is where caffeine is present in a plant. Why do plants produce caffeine? Well there are all types of reasons from pollinator attraction to pest deterrence, but in the case of the tea plant, it's pest deterrence. One of the most important parts of the plant is the young, fresh growth. This is what the tea plant wants to protect from pests and therefore has a high concentration of caffeine in the buds compared to lower leaves.

Now that we understand these principles, we can better understand whether White Tea does in fact have less caffeine than a Black Tea for example. As I said above, the simple answer is no, but it is much more complicated than that. It depends on cultivar (type of tea plant), picking grade (whether it was a bud heavy White Tea like Silver Needle or a leaf heavy White Tea like Shou Mei), etc. There is one thing that does give us a bit more of a helpful answer however.
White Tea is a simple way of making tea - the leaves are picked and then dried, that's it. Black Tea on the other hand goes through many different processing steps, one of which being a cooking stage. As noted above, L-Theanine diminishes with heat and therefore is less present in Black Teas than White Teas. What does this mean when it the tea is steeped and ready to be drank? Regardless of caffeine level, White Tea usually feels more calming. There's a reason that White Tea is usually drank later in the day whereas Black Tea (or better yet, matcha) is drank early in the morning. It's not necessarily because of caffeine levels, but more specifically the levels of L-Theanine present in the tea and therefore the amount of caffeine that the drinker really feels.
If you want to try a smooth and calming White Tea, we've got in a small batch rustic tea from Sri Lanka that we've called Flux. It is handmade Shou Mei style White Tea with a lot of depth and a calming effect.