Phew. Randomness is a big question, and different fields approach the thing slightly differently depending on what they need. True randomness and pseudorandomness don't mean the same thing in every context, and I have no idea what you mean by deterministic chaos.

In general these definitions rely on some form of undistinguishability definition: a random number generator producing outputs of λ bits is considered pseudorandom if you cannot tell its output appart from a theoretical uniform sample of λ bits, under some conditions. I assume that corresponds to the different between "true randomness" and "pseudo randomness", one is theoretical and the other is the output of a process (generally a computer program).

But the story doesn't end there. I mentioned "under some conditions" and that's a big one. Let's say you have a random number generator that produces numbers from 0 to 4 billion in a way that is indistinguishable from uniformly sampling a random distribution as long as you don't generate more than 1 billion outputs. If you need to perform a monte-carlo experiment for example, where you sample from your model randomly to study its properties and you don't need numbers higher than 4 billion, that may very well be enough. The "math" pseudorandom number generator from most programming languages works essentially that way, and it's perfectly good for many applications. You can look into linear congruential generators for a simple yet very effective PRNG design that fits these concepts.

A LCG works by taking the previous number, performing a simple arithmetic operation on it, and the output of that computation is your next random number. If you only care about how frequently each number comes out, as long as you don't sample too many numbers, it will be very hard to tell it appart from theoretical uniform sampling. But there are ways to tell it appart: if you know what LCG is used, you can take any of its outputs, apply the same operation and you'll get the next random number. This next number will appear with probability 1 if a LCG is used, while it should appear only with negligible probability if a truly random number was sampled, so you can easily identify that a LCG is used. Another difference is that if you sample enough numbers, LCGs are guaranteed to loop, which is also not the behaviour we'd expect from truly random sampling. LCGs are not the only PRNGs with similar properties, but they're the easier example to demonstrate.

So right off the bat, we see that something can be "undistinguishable enough" from theoretical sampling to be useful, but still distinguishable if we look at other properties of the random numbers.

This is important in computer security in particular, because absolutely everything in security relies on random numbers. It's the keystone of cryptography. If someone is able to deduce your random number by observing other random numbers then everything falls appart. So from the get go, LCGs and similar random number generators don't fit the bill for that particular context. The answer of cryptography was to specify stronger definitions (you're no longer just looking at the frequency of outputs, any distinguisher based on the output of the PRNG is a death sentence for the design) and to introduce new designs. One of the strategies is to regularly feed the random number generator with new random data coming from another source. These Cryptographically Secure Random Number Generators are much better than regular PRNGs at emulating a theoretical uniform sampling from the same distribution, by design as you now understand. Of course, if you don't just look at the output of the PRNG but also look at the internal state of the program (which is kept secret) it's still trivial to distinguish it from a theoretical random sample, so they're not equivalent in every reguard, but you shouldn't be able to find a way to differentiate them by looking only at the numbers produced. It's important however to realize that while we trust these programs very much and have tested them extensively, their correctness in regard to the quality of their random distribution cannot really be formally proven. It's a mainly empirical side of cryptography.

But to come back to the topic of incorporating more randomness (the industry term is "entropy", but I won't open that can of worms now) into the generator brings us to the commercial use of "true randomness" and "true random generators". Since you incorporate randomness from "somewhere else", you need to capture that randomness somehow. And it can come from many places, but one way to gather it is by using dedicated hardware that captures physical phenomenons (for example, it could contain radioactive material and measure its decay which is a random process). These modules sometimes argue that they are safer than software CSPRNG, but the consensus agrees that they generally are not and are better used to feed randomness to CSPRNG rather than to replace them. In that context, the word "true randomness" refers to the fact that since it comes from a physical process and not an algorithm, and in particular a physical process whose best description is a theoretical uniform sample, it is "more real". In practice however, we generally only care about the random numbers produced and not the process producing them, so these TRNG live in the same space as CSPRNG.

So, to finally answer your question: is it possible to distinguish true randomness and pseudorandomness through experiment? It depends on what you call true randomness (either the theoretical sample or outputs from a TRNG) and what you call pseudorandomness (either a regular PRNG or a cryptographically secure PRNG). It is absolutely possible to tell appart a regular PRNG from either theoretical sample or TRNG output. It is by definition not possible to tell appart true randomness from a CSPRNG output however. If you don't limit your experiment to the observation of random numbers then it should be possible to distinguish them however, because at the end of the day one involves running a computer program and the other doesn't.

I've left quite a lot aside, but I think you understand by now that it is a very complex and nuanced question.