Okay, let's think about it, shall we?
You already see both distributions. But let's assume that you don't. That is, you know nothing about the type of the distribution, whether it is random or otherwise. But you know that any random distribution is, well, random, that any pattern you might look for would also be random. However, you also know that with a random distribution you are bound to find some patterns, and this is not "random" specifically because it is a random distribution
So how random is it really? If you see a dot, aren't you more likely to see another dot nearby with such a distribution? But that means things are no longer random to you even if the distribution of dots itself remains totally random. You take advantage of some feature or property of a random distribution that any random distribution has (namely, patterns), and thereby you stop it being random despite it being random. Isn't it a nice paradox or conundrum?
There is no such thing as "random to you", at least there is no use for such a thing unless you're into some weird art forms. For any practical use of randomness, such as gambling or cryptography or statistics, math trumps human perception of randomness.
You seem to be basing your opinion on an axiom that random has patterns but it doesn't. Despite the appearance of patterns (alltho I can't say that I see any patterns in your second picture - a few dots next to each other is not a pattern) there are no proven patterns in, for example, bitcoin RNG. Which has a big big incentive to be cracked, wouldn't you agree? So random is random.
So how random is it really? If you see a dot, aren't you more likely to see another dot nearby with such a distribution?
In a truly random distribution you should expect another dot anywhere with an equal chance, including next to the first dot. It is not more or less likely. It's an optical illusion. On the contrary, if no dots at all have another one nearby that is definitely not random.
For example in a 100x100 picture if you have 100 pixels on it then the next pixel has a ~ 1/25 chance (maybe slightly less due to edges etc) of touching another one and it gets much higher as you put more of them down.
Here's another one. If run a random "pixelator" enough times you should have a non-zero chance of creating a picture where all pixels are in a single large block (similar to a winning/losing streak in gambling).