I can see why you're confused, Majka. In a way, all three statements are true. This is why:
1. "The only correct possibility is "I like reading".
I can understand why you were told this, especially if the teacher was working from an older British textbook. In the past, in British English, the verbs like, love and so on were always followed by the gerund ( V+ ing form). That's what I would always say, and that's what most British people say. I remember, many years ago, noticing that speakers of American English would say "I like to read", and it struck me as strange - I had never heard this construction before. However, language changes, and both constructions are now widely used.
2. "There is a slight difference in meaning".
Yes, there is - at least for some speakers. The statement "I like reading" means that I enjoy reading : in other words, the act of reading is pleasurable for me. The statement "I like to read" can have the same meaning as "I like reading" - especially for American English speakers - but it can also have a slightly different meaning. If you say, for example, "I like to read a page of my Russian grammar book every morning before breakfast", this may have nothing to do with pleasure. It means that I think that it is a good thing to do. The difference is more obvious when you say something like "I like to have a dental check-up once every 3 months" - this clearly isn't an enjoyable experience in itself, but it's something that you like to do because it's good for you.
3. "The meaning is the same".
Yes, this is often true. As I mentioned in (1) , both constructions are now common worldwide. So, if you ask a group of native-speaker book-lovers about their free time, some will say "I like reading", and others will say "I like to read", and the meaning would be the same.