Thank you for the answers guys! So, BIP 47 just makes blockchain analyzers work a bit more difficult. Is it that we have to have both - unlinkability and untraceability - to be anonymous? I mean, why?
Honestly, I don't understand this.
I think unlinkability is not being able to link past payments, whereas untracability is not being able to to trace future payments. Please someone correct me if that's wrong though.
That is part of it yes. Without untracability anyone you pay can follow the trail back and unambiguously identify your other associated transactions (including those that paid you). Without unlinkability once you pay someone you can see other payments (past or future) to the same recipient. The other aspect of it is that a third party can see both of these types of connections, which are the building blocks of blockchain analysis.
"To be anonymous" is a bit vague since it means different things in different contexts. If you want a coin where the chain is hard to analyze and in doing so identify people with transactions, then you want to break both sorts of links described above. Hiding amounts is helpful too.
I like to compare it to, say, mailing cash to someone. Even though the sender knows your address, he can't see what, if any, other payments you have received (ditto for an outside observer). That's unlinkability. No one except you and the sender can tell what amount was sent (confidential transactions).
Finally, assume no return label: even you the recipient has no idea where the cash came (much less an observer). That's untraceability.
The analogy somewhat holds for an in person exchange as well, but it's much harder to be first hop anonymous in that case.