In advaita vedanta, consciousness is the self.
In Buddhism, consciousness is consciousness because it cognises (MN 43). To quote MN 43:
'Consciousness, consciousness': Thus is it said. To what extent, friend, is it said to be 'consciousness'?
'It cognizes, it cognizes': Thus, friend, it is said to be 'consciousness.' And what does it cognize? It cognizes 'pleasant.' It
cognizes 'painful.' It cognizes 'neither painful nor pleasant.' 'It
cognizes, it cognizes': Thus it is said to be 'consciousness.'
Consciousness has 2 modes: it can be pure consciousness, and it is described as "being aware of being aware"
Yes, consciousness knows consciousness. MN 43 says:
Yā cāvuso, paññā yañca viññāṇaṃ — imesaṃ dhammānaṃ saṃsaṭṭhānaṃ no visaṃsaṭṭhānaṃ paññā bhāvetabbā, viññāṇaṃ pariññeyyaṃ.
Discernment & consciousness, friend: Of these qualities that are conjoined, not disjoined, discernment is to be developed,
consciousness is to be fully comprehended.
PTS Pali English Dictionary pariññeyya adjective knowable,
perceivable, to be known (accurately)
What is "perceivable" cannot occur without consciousness. To quote MN 43, again:
Feeling, perception, & consciousness are conjoined, friend, not disjoined. It is not possible, having separated them one from another, to delineate the difference among them. For what one feels, that one perceives. What one perceives, that one cognizes.
or the "I that I am knows that I am";
No. The above is thinking or 'sankhara'. Buddhism teaches about five distinct aggregates that compose of life (SN 22.48). These are physicality, feeling, perception, formations (sankhara) and consciousness. Any type of label, word, thought or description is a formation (sankhara); particularly the delusion of "I am" (refer to SN 22.81).
while the other mode of consciousness is consciousness entangled in objects, like feelings and thoughts.
No. Consciousness does not "entangle" itself. What is entanglement is "sankhara" ("thinking wrongly").
Is pure consciousness, or being aware of being aware, the same as what the Buddha called the "unconditioned"?
No. Consciousness is a mental phenomena, that arises dependent upon conditions (MN 38). The unconditioned is Nibbana, which is not a mental phenomena (MN 115).
Or is the unconditioned the cessation of consciousness, as Buddhism sees consciousness as impermanent and not self?
No. Cessation does not mean cessation of consciousness. It means cessation of ignorance that causes consciousness to get wrongly involved in unwholesome sense objects. It means the cessation of a consciousness tainted and enslaved by ignorance.
They said if consciousness was not aware of itself, none of us would know that we are aware.
Sure. So what?
I have this problem, my friend had the same, so we wanted to ask you if you could help us out.
This problem appears to be a strange attachment & obsession with useless 'philosophy'. Buddhism is about ending suffering rather than philosophizing.