To say that one has taken somebody's advice will normally be understood to mean that one has not only heard, understood, and considered the advice, but that, upon considering it, one has determined that the advice is well justified and worthy of being followed. In other words, taking the advice involves embracing the views expressed in the advice as one's own. One can thus say, 'X, Y, and Z had given me different advice on the matter; after thinking about it for a while, I took Y's advice and did this'. In that situation, in which one has rejected X's and Z's advice, it would be misleading to say that one has taken the advice of all three of them, even though it would be true that one has received advice from all of them.
Saying that one has taken somebody's advice still does not, strictly speaking, imply that the advice has already been followed: it may be that the time to implement the advice is still in the future. What it does imply is the intention to act on the advice when the time for so acting arrives. One can say 'I took your advice to go on that trip, and started planning for it, but then got sick and had to cancel it'; in that case the advice to go on the trip hasn't been actually followed, but it is still true that one took the advice in that one formed the intention to follow it.
Sometimes one forms an intention to follow somebody's advice, whatever it turns out to be, before soliciting it. In these scenarios, obtaining the advice merges with taking it, in the above explained sense, which explains why take advice may also be understood to stand for obtaining advice. The authors of the quoted dictionary definition probably had such scenarios in mind (note that they say 'typically from an expert', and that seeking expert advice usually involves an intention to follow it).
The explanation offered in the preceding paragraphs does not apply to constructions such as take somebody's advice seriously; it is not self-contradictory to say 'I took your advice seriously, but decided to reject it, because it was outweighed by other considerations'. In that sentence, seriously doesn't qualify take somebody's advice in a straightforward manner; it is rather that take seriously functions as an idiom.