Anacrusis (pickup) is a bit more rhythmic than melodic. Hearing it seems easy to my musical brain, but I can understand how it would not be easy for others. Most music has a set rhythm, which we can understand in its simplest form by saying there is a fairly low number (most commonly 4), to which one can repeatedly count while listening to a piece of music, such that the piece will seem to flow with the counting.
WhatWait, what does that even mean? Let's use Happy Birthday as an example. If you count "one two three one two three one two three" repeatedly while someone is singing Happy Birthday, you'll hear/feel that the song seems to flow along with your counting, except you can't start the song with "one" right when the singer first says "Happy-". You have to start your count on "three" and then go back to "one", with the "one" matching when the word "Birthday" is sung. That's anacrusis. Here's a chart-like thing to hopefully make it more visual:
So by now you're probably ready to ask, "Wait, why don't we just make 'one' fall on 'Happy' and then 'two' for 'birthday' and so on?" Excellent question, and this is the very core of anacrusis, and even rhythm in general. How do we know that it's a three-count? How do we know where the 'one' is?
That last phrase in step three is really what the articles you read seem to be about. Anacrusis could be in the eye of the beholder. Going back to the question of why we don't just count "one" on "Happy" in Happy Birthday, we could actually count "one" on "Happy" and then count sixes or twelves or even fours (which would create a different feel, sometimes called "three over four") and then we would not consider there to be anacrusis for Happy Birthday. Anacrusis and the rhythms we typically apply to music are merely conventions of western music theory, and other cultures can and certainly do develop theories and musical ideas that don't match, or even outright clash with the concepts that have their origins in western Europe and even back to ancient Greece.