Is this a right explanation of phenomena of enlightenment by Osho?

Personally, I don’t take ‘enlightenment’ to be a fixed destination, but rather, it is a continuing journey, so Osho’s definition is true, but for me, incomplete.

For example, in the Surangama Sutra, the meditation path of Avalokitasvara has 14 progressive stages, the very first of which corresponds to Osho’s ‘enlightened’ state:

First, because I did not listen to sounds and instead contemplated the listener within, I can now hear the cries of suffering beings throughout the ten directions, and I can bring about their liberation.

The bit you quoted reminds of the chapter titled The benefits of the Kusulu's meditation and the few chapters before that, in the book Vivid awareness by Khenchen Thrangu.

It says that (I paraphrase)

• The nature of the mind is to know [perhaps that's the same as "witness"]
• Realising the nature of the mind is what's called "transcendant intelligence" or "great mother" in the second turning of the wheel
• That's related to buddha-nature [and so, "enlightenment"] in the third

"Actions" are mentioned seven chapters earlier.

In between, it mentions that:

• Two types of meditation (i.e. tranquility and insight) are taught
• The clarity needed for insight [into the nature of the mind] depends on tranquility
• This teacher recommends they be practiced together.

And example of "tranquility" is "resting peacefully free of thoughts within the empty aspect of the mind" [which is like the quote's saying "nothing else to witness"], but that "we do not really recognise the mind" which is what makes that tranquility without "insight" -- clarity and emptiness are both important to budda-nature.

Perhaps you can see parallels (between this and Osho's teaching).

However I don't know about the first quote i.e. "... conscious of itself". The book I'm referring to seems to talk about consciousness of "emptiness" or of "clarity", but not IMO of "self" or of "self-consiousness"; which ("self") I think of as a "tangle" and one of the primary differentiators between the Buddhist and Hindu traditions -- with Hinduism teaching Atman, maybe contradicted by the Buddhist doctrine of Anatta.

Perhaps someone could say they're "ultimately" the same or similar. I guess I find the Buddhist doctrine easier or more "liberating". Like Occam's razor ("prefer the simplest explanation" and "shave away unecessary assumptions"), I find the doctrine easier without any references to "self".

Although there are many points in common between Osho's teaching and the Buddha's, there are some subtle differences. In particular, Osho talks about self-realization, while the Buddha addresses the end of suffering:

MN10:2.1: “Mendicants, the four kinds of mindfulness meditation are the path to convergence. They are in order to purify sentient beings, to get past sorrow and crying, to make an end of pain and sadness, to end the cycle of suffering, and to realize extinguishment.
MN10:3.1: What four?
MN10:3.2: It’s when a mendicant meditates by observing an aspect of the body—keen, aware, and mindful, rid of desire and aversion for the world.
MN10:3.3: They meditate observing an aspect of feelings—keen, aware, and mindful, rid of desire and aversion for the world.
MN10:3.4: They meditate observing an aspect of the mind—keen, aware, and mindful, rid of desire and aversion for the world.
MN10:3.5: They meditate observing an aspect of principles—keen, aware, and mindful, rid of desire and aversion for the world.

The Buddha also gives a clear warning about wrong immersion. Not all meditation is right. Some meditation is wrong:

AN10.119:13.1: ‘Wrong immersion has a bad result
AN10.119:13.2: in both this life and the next.’

I learned a lot reading Osho's books, but I had questions left unanswered until I read the suttas. And one sutta in particular answered a lot of my questions. In MN8, the Buddha talks about self-effacement, not self-realization. In MN8, the Buddha explains the importance of ethics:

MN8:10.1: It’s possible that some mendicant, going totally beyond the dimension of infinite consciousness, aware that ‘there is nothing at all’, might enter and remain in the dimension of nothingness.
MN8:10.2: They might think
MN8:10.3: they’re practicing self-effacement.
MN8:10.4: But in the training of the Noble One these are not called ‘self-effacement’;
MN8:10.5: they’re called ‘peaceful meditations’.

And in his teaching to Venerable Mahācunda, the Buddha simply says:

MN8:12.2: ‘Others will be cruel, but here we will not be cruel.’
...

Right immersion requires a basis in ethics, and Buddhist ethics are very subtle, deserving careful study. Without ethics, we fall into wrong immersion, wrong knowledge and wrong freedom.

MN8:12.19: ‘Others will have wrong immersion, but here we will have right immersion.’
MN8:12.20: ‘Others will have wrong knowledge, but here we will have right knowledge.’
MN8:12.21: ‘Others will have wrong freedom, but here we will have right freedom.’

Perhaps the question we might ask ourselves is not "is this self-realization?". Perhaps we might better ask "is this right immersion?" And to answer THAT question, we would need to examine our entire practice for right view, right thought, right speech, right action, right livelihood, right effort, right mindfulness, right immersion and right knowledge.

A few points that occur to me...

First, when perception (witnessing, seeing, noting or noticing, awareness, etc) has an object there's still a duality, a mind-iness, a separation of this from that. Those separations are where dukkha and tanhā grow. The passage:

Your whole witnessing jumps upon itself. It witnesses itself, because there is nothing else to witness.

...is a self-reference, and so it hasn't quite eliminated that objectification. We can even sense that residual mind-iness in the last phrase, where we have corralled the mind into taking a self-referential object, but it's still the mind taking. Perhaps it's just bad phrasing on Osho's part, but...

Second, Osho's description sounds a bit solipsistic (or perhaps narcissistic) to my ear. He puts more emphasis on bliss than I would; he seems to suggest that we should shut out the world until we are left with ourselves alone and the pure joy that entails. As presented here, it feels like a heating contraction into the self, not a cooling expansion into the world. There are different styles of practice and practitioners with different temperaments, so this may work for some. But ultimately I think one must align the world and the self, and I don't see how this gets one there.

Third — and this point may be a bit abstruse — mediation is merely a practice, and skill in the practice of meditation is not the same as realization. Realization is an attitude, in the deep, philosophical sense of the term 'attitude': a way of holding oneself within the world with perfect attention and skill. We practice that attitude on the cushion — just as an athlete might practice the attitudes of his sport at a gym — but what happens on the cushion is ultimately as inconsequential as what happens at a gym. I have no doubts that Osho's practice will take one as far as he claims; it's good practice as far as it goes. But what he holds here as the 'ultimate experience of bliss' isn't the ultimate aim.

Osho's statement quoted by OP:

The mind is empty, the heart is empty, the body is relaxed. In that moment happens something like a quantum leap. Your whole witnessing jumps upon itself. It witnesses itself, because there is nothing else to witness.

The statement above sounds a bit like the jhana levels, especially the fourth jhana level according to the sutta quote below.

However, it doesn't sound anywhere close to Nibbana and full liberation from suffering.

This is not unusual as many self-proclaimed gurus in modern times (perhaps like Eckhart Tolle), in the past (perhaps like Osho) and also in the time of the Buddha (like Alara Kalama), have apparently attained various levels of jhana, but had not necessarily attained Nibbana. Of course, I can't say this with certainty, but it would be a reasonable speculation.

Even if those teachers are not completely liberated, I'm sure they have something useful to teach to students who are not as advanced as them.

"And furthermore, with the abandoning of pleasure and stress — as with the earlier disappearance of elation and distress — he enters and remains in the fourth jhana: purity of equanimity and mindfulness, neither-pleasure-nor-pain. He sits, permeating the body with a pure, bright awareness, so that there is nothing of his entire body unpervaded by pure, bright awareness.

"Just as if a man were sitting wrapped from head to foot with a white cloth so that there would be no part of his body to which the white cloth did not extend; even so, the monk sits, permeating his body with a pure, bright awareness. There is nothing of his entire body unpervaded by pure, bright awareness.
AN 5.28