Perhaps my experience will help someone in their future rebookings and dealings with LH. Here is the story:
I had a ticket: MAD-ZRH-GRU return in P. Called in on 07 July to rebook it to PMI-ZRH-GRU. LH rebooked and took my credit card to cover 20 EUR tax difference (this amount was blocked on my CC). The agent said that the ticket should be reissued in 2 days max. Of course, nothing was re-iussed in two days and there I went calling daily and getting a standard "it is in urgent queue".

On 16 August PMI-ZRH gets cancelled, so I call LH and get rebooked to PMI-GVA-ZRH. The agent says it's free because it's involuntary and sends to re-issuing. The ticket does not get reissued, either. Two days later I called back to reroute for PMI-FRA-ZRH (given that PMI-GVA-ZRH-GRU has not been re-iussed). The agent quotes fare difference of US$3,000. I mentioned that this would be involuntary rebooking due to cancelled PMI-ZRH and not re-issued PMI-GVA-ZRH-GRU. And then we go:
1) first agent: yes, we can rebook via FRA for free. She does so and sends to ticketing (nothing get re-ticketed)
2) second agent: yes, we can rebook via FRA, but it will be 30 EUR. I gave my credit card details and three days later (another amount blocked) I get call from LH saying that they made an error in calculation and it is 46 EUR. OK, I agreed. Two days later - another call with a voice mail saying that they made an error and the new difference is 1450 EUR. I don't take any action.
3) third agent: no, we cannot rebook for free via FRA but we can rebook via ZRH (with a 26 hour layover) for free.
4) fourth agent: we cannot rebook for free because your original request to change MAD-GRU to PMI-GRU was never re-issued, so the cancellation of PMI flights does not allow you to ask for free rebookings afterwards.
5) fifth agent: yes, we can rebook, but there are no seats in C (on lh.com plenty are sold)
6) six agent: same as 4th. But this agent was super helpful and tried several options. At the the end, she understood the entire story how LH took credit card details several times but never charged the card, how various price differences were mentioned within the same hour. She then came back saying that unfortunately not free rebooking but the supervisor authorized full refund, if I was interested. So, I took it and booked with AF for slightly more.

I understand that agent 4 is right: as the new ticket was never re-issued, I had no right for involuntary rebooking. What bothers me is that after I called to change MAD-GRU to PMI-GRU and gave me credit card, I, as a pax, should not baby sit the ticket. The transaction went through (amount blocked on my credit card) and the agent said: it's confirmed and now sent for re-issue. Most people would understand it as: OK, I am done so now I can adjust the rest of my trip...

Any thoughts what I should have done differently (apart from calling hourly)? Thank you.