Hi Cecilia,
Thank you for your reply! I actually solved it, and it's a curious one.
So, yesterday I solved it while talking to customer service and I thought what they did (resetting SIM settings) was the solution. After few hours, however, the problem was still there.
I did some further tests, and here is how I solved it - sharing it so maybe someone with the same problem can use this:
(1) Time and date setting needs to be automatic, not manual. [Right now I am using the SIM in an old phone, not a smartphone, and when you remove the battery then you can set up date and time manually. Don't do it, go to the settings and have it done automatically.
(2) Right now, only one provider (in my case Airtel) will let me get the messages. If I switch back to Vodafone or to any other provider, it simply doesn't work. Airtel is not always available, but it's the only one that works. I think it's very strange that Vodafone doesn't work though.