In the second example the first "that" is a conjunction (or a conjunctive participle.) The OED explains how it is a conjunction:
I. 1.a. Introducing a subordinate noun clause, as subject, object, or other element of the principal clause, or as complement of a n. or adj., or in apposition to a n. therein. The subordinate clause as subject is most commonly placed after the verb and introduced by a preceding it, e.g. ‘it is certain that he was there’ = ‘that he was there, is certain’. As object, it usually follows, e.g. ‘I have heard that he was there’.
[This use of that is generally held to have arisen out of the demonstrative pronoun pointing to the clause which it introduces. Compare
(1) He once lived here: we all know thát;
(2) That (now this) we all know: he once lived here;
(3) We all know that (or this): he once lived here;
(4) We all know thăt he once lived here;
(5) We all know he once lived here.
In senses 1 – 3 that is a demonstrative pronoun in apposition to the statement ‘he once lived here’; in sense 4 it has sunk into a conjunctive particle, and (like the relative pronoun) has become stressless; in sense 5 it has disappeared, and ‘he once lived here’ appears as the direct object of ‘we know’.
After aware, certain, conscious, suspicious, assured, informed, persuaded, etc., of or some other preposition seems understood before that:
‘I am certain of that: he once lived here’. But ‘I am certain that’ may have arisen as another way of saying ‘I know that’; and so of the other expressions.]
1873 J. Morley Rousseau I. vii. 284 Rousseau was persuaded that Madame d'Epinay was his betrayer.
“While some might think that all professors do is lecture a few times a week and assign some homework, that is not all, Prof”.
As you see, this falls into (4) above, and can be omitted as per (5).
The second “that” is a pronoun with the referent of “what I have just said”, or the whole of the previous clause.