A simple explanation of a flaw that makes no reference to Lob's Theorem, meta-anythings, or anything complicated. Of course, spoilers.

"Let ◻Z stand for the proposition "Z is provable". Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C."

This statement is the source of the problem. For ease of typing, I'm going to use C' = We can prove C. What you have here is (C'->C)->C'. Using Material Implication we replace all structures of the from A->B with ~A or B (~ = negation). This gives:

~(~C' or C) or C`

Using DeMorgan's laws we have ~(A or B) = ~A and ~B, yielding:

(C' and ~C) or C`

Thus the statement (C' -> C) -> C' evaluates to true ONLY when C' is true. You then proceed to try and apply it where C' is false. In other words, you have a false premise. Either you can in fact prove that 2=1, or it is not in fact the case that (C' -> C) -> C' when C is "2=1".

PS: I didn't actually need to read Lob's Theorem or even know what it was about to find this flaw. I suspect the passage quoted is in fact not the result of Lob's Theorem. You can probably dig into Lob's Theorem to pinpoint why it is not the result, but meh.