SuttaCentral
“Regarding this, Master Gotama, a demonstration of psychic power is experienced only by the one who performs it, occurring only to them. This seems to me like a magic trick."
What is meant by this part?
For example: “flying cross-legged through the sky like a bird”. I would think that another person can see him flying through the sky if he does this. So it occurs to everyone watching?
I read “experienced only by the one who performs it” as meaning only that one person has the experience of actually doing it. Only that person knows how to do it.
Likewise…
" comprehending the mind of another person…
is also experienced only by the one who performs it, occurring only to them."
(AN 3.60 Sc 15.5 - 15.7)
So after a person demonstrates a power or comprehends the mind of another, they are still the only one who knows how to do it.
But the third kind of demonstration, where someone instructs:
“But as to the demonstration where someone instructs others: ‘Think like this, not like that. Focus your mind like this, not like that. Give up this, and live having achieved that.’
I prefer this demonstration, Master Gotama. It’s the finest of the three kinds of demonstration.”
(AN 3.60 Sc 16.1 - 16.3)
If your demonstration is instruction, everyone leaves able to experience what was taught.
IMO, the psychic powers such as flying cross-legged, walking on water, diving into the earth etc are done using the so-called ‘astral’ body, not the physical body. You’ll note that the psychic power pericope always come after the mind-made body pericope, which for me refers to the astral body. In the OBE (out-of-body experiences, also called astral projection) literature (Robert Monroe ect), there are plenty of testimonies of such experiences (flying, going through walls etc) and many comments about not being recognized by physical people. Some says they their cats could see them though! 
quite literally, here:
touching and stroking with the hand the sun and moon, so mighty and powerful
SN51.11 for example
If you accept the modern scientific understanding of what and whereabouts the sun and the moon are, is this to mean that they “fly cross-legged through the air like a bird”, leave the atmosphere of the Earth, and travel 100 million miles through the vacuum of space to “touch the [massive violent glowing ball of nuclear fusion that is the] sun with their hand”. I also read most of the ‘psychic powers’ as basically entering into a dream-like state. All of these things are possible in dreams, and if one enters dreaming lucidly can be “controlled”.
Of course, as @JimInBC mentions, something like mind-reading seems more demonstrable in waking life. Mentalists can demonstrate seemingly miraculous feats of mind-reading without ever having trained in the Dhamma-Vinaya, some of them maintain the air of mystery associated with stage magic, but that’s undermined by the rest who simply state that it is trickery based on psychological understanding of conditions/conditioning, probability, and reading non-verbal cues.
Haha, took me a while to get what you meant 
Yes, but in a lucid dream, from what I understand, your environment is still your own creation, and cannot be shared with another person, it’s a reality which is solely experienced and controlled by the (lucid) dreamer. While in astral projection, one experiences one of the astral worlds/realms, which are a common reality for all the beings that inhabit or visit them (like the physical world is a common reality for physical beings).
I wonder about that one yes. Could it be maybe an idiom? For example meaning ‘geting very close to the sun and the moon’, rather than literally stroke them with one’s hand? I read accounts of astral projectors who visited the moon but none of them visited the sun, although one said that he flew through it…
Agree, once one believe in the reality of mind reading (and thought implantation), a lot of magician and mentalist (e.g. Lior Suchard) ‘tricks’ (which are not tricks in this case!) are much easier to explain. And much more believable that some of the explanations that skeptics try to come up with to debunk these so-called tricks.
The standard psychic power passage of the suttas reads like the speaker is saying: No really! I go there with the body, not just in my mind! I literally touched sun and moon.
Remember how in SN 51.22 Ananda asks (as a proxy for the audience): Do you mean it metaphorically (then it wouldn’t be a superpower) or with the physical body composed of the physical elements? And the Buddha confirms the latter.
Turns out, with our scientific understanding today, that exactly the emphasis of the text which is supposed to impress the audience with its bold statement, makes it literally impossible that this was a physical superpower. A ‘real’ traveler would have added: "Btw, outside of earth there is no air, so I had to hold the breath for a very long time. Also, the sun is unbelievably hot, in order to touch it I had to protect the body with an additional superpower of immunity, so the casual meditator shouldn’t just try it’…
Well, assuming the existence of an astral body and astral realms, touching the astral moon and sun with one’s astral body is as literal as doing it in the physical world. The literal character of the passage does not indicate that it is done in the physical world, only that it is real, and both physical and astral worlds are real (again, assuming their existence, which is a matter of personal belief or experience).
In this sutta the Buddha confirms that he can visit the Brahmā world either using the mind-made body (i.e. astral projection IMO) or the physical body.
In response to Ananda’s amazement, the Buddha replies that the Realized Ones have amazing qualities. I understand this as meaning that only Buddhas can do these things. Since the psychic powers can be achieved by some meditators, I conclude that Buddhas and advanced meditators can attain the Brahma realm using the mind-made body while only the Buddhas can also use their (“lighter, softer, more workable, and more radiant”) physical body.
But maybe other advance meditators can also use their physical bodies… maybe it’s just a more difficult and therefore a rarer thing… What’s your take on this?
I think I’m with you in the interpretation that it could be an astral body, but it could just as well be a fancy nimitta (in the commentarial sense). But that isn’t the implication of the passage, isn’t it? The text teases an audience which eagerly wants to believe that the Buddha had miracle-like powers, not ‘just’ astral ones. When I hear ‘travel through space like a bird’ then I as a listener am not invited to think of an astral body - I think of a bird, then imagine me flying like a bird, then realizing that that’s impossible and then I’m amazed that the Buddha can do the impossible.
It would be a very different story if the text said “what seems like a superpower to you is based on the production of a mind-made body. This body has no obstacles like the physical body made of the four great elements. It can go unhindered through a wall, fly like a bird, touch the sun…”. Because then the audience would be like “Aaah, it’s not a miracle, it’s a science”.
I wouldn’t know if the whole passage is made up, or if the astral/nimitta aspect was purposefully omitted with time. But I smell the intent to wow the audience, and that in some way or another a distortion took place.
For me, reading or listening to someone who can fly with an astral body is already a lot of wow! But maybe I’m more easily impressed than the folks of yore 
But I see what you mean, could well be that indeed.