I think that Tom's envelope to the doctor was a gratitude for a job well done, similar to a tip to a waiter, though a little demeaning. It's always better to ask "how much do I owe you?" A bribe is paid before the service in order to get a thing done! Also, I believe that paying bribes is ingrained in the population by the previous regime. We have a friend who yearly "buys" a spa "poukaz" which she really doesn't need. Talk about a habit!
We have also lived for three years in Slovakia. We have never paid any bribes to anybody, though it was suggested by some of our friends that we probably should, and that includes traffic stops, etc. Everything that we needed was executed in the course of normal official interaction, without any preferential treatment, etc., basically to our satisfaction. Isn't it the expectation that "you have to pay" which encourages bribery?