That's an interesting question, and I hope you'll bear with me while I try to answer it.
Normally, if music is written with two or more staves, it's understood that each staff represents a different part-- i.e., different instruments (e.g., violin vs. trumpet), or perhaps different people playing the same type of instrument (e.g., 1st violin vs. 2nd violin).
When the music is for two or more different parts, the staves for the parts are joined together on the left with a straight line-- "|"-- so it's understood that they're meant to be played together at the same time.
In contrast, when the music has multiple staves that aren't joined together by a straight line, it's understood that they're played at different times-- normally one following the other, in sequence, like reading lines of text on a page. (However, special symbols or abbreviations might be used to indicate where they're to be played in some other order-- e.g., a repeated passage with two or more different endings, or that jumps to a coda the last time through, etc.
The first version you posted of "Take On Me" is an example of this-- there are multiple staves on the page, but none of them are joined together on the left by a straight line, so they represent the lines of a single part and are played one after the other in sequence.
On the other hand, if two staves are joined together on the left by a pointy bracket-- "{"-- it's understood that they represent a single part or instrument rather than two different parts. Usually the part is for a piano or other keyboard instrument that has a range of notes too wide to be comfortably written using just one staff. And usually (but not always) each staff represents a part played by a different hand-- i.e., the lower staff is for the left hand, and the upper staff is for the right hand. In that case the left-hand part is usually written with a bass clef, and the right-hand part with a treble clef, but this depends on the notes themselves. Sometimes a bass clef will appear in the right-hand part, or a treble clef in the left-hand part. And sometimes which hands are used to play the notes has nothing to do with the upper and lower staves per se-- e.g., if there's a run of notes that begins at the far left and goes up the keyboard to end at the far right, then the notes will probably have to be played by alternating hands-- left, right, left, right, etc.-- such that some of the notes in the lower staff must be played with the right hand, and some in the upper staff with the left hand.
The second version of "Take On Me" is an example of a more complex arrangement-- there are multiple staves on the page, but some of them are grouped together by a straight line, with multiple groupings on the page. So each grouping of staves represents parts played together at the same time, and each new occurrence of grouped staves indicates a continuation from the line immediately above. And to complicate matters, two of the three staves in each grouping are joined together by a pointy bracket. So even though there are three staves in each grouping, there are actually only two parts-- probably a keyboard part and a bass guitar part (judging by the clefs used). Of course, there are also chord names written between the lower staff of the keyboard part and the staff for the bass part, so you could say that the chord names are for a third part-- namely, the guitar. And there are also lyrics, which represent a vocal part. But the three staves in each group represent two parts, not three.
Whew! To get back to your question... Wait, what was your question again? (Just kidding!) When there are multiple staves, but either one person isn't meant to play all of them, or else you want to play the keyboard part but don't want to play both staves of that part, then you can decide which staff or staves you do want to play-- unless you're playing with other people, in which case each person will be assigned which staff or staves they're expected to play. But assuming you're playing by yourself, and are trying to decide which staff to play with your right hand (while your left hand plays chords), you might want to play the part that goes with the lyrics or main melody. That's assuming you aren't singing the lyrics while you play, because if you are singing then you might want to play some other part with your right hand. Unfortunately, sheet music like the second version of "Take On Me" can make that more complicated, because the upper staff of the keyboard part includes an instrumental part (e.g., the first three-and-a-half measures) mixed in with a vocal part, so what are you supposed to do then? In that case, you can play the introductory instrumental part, then jump to a different clef of your choice for your right hand while you sing the main melody-- e.g., maybe you could play chords with your right hand and play the bass guitar line with your left hand, or something like that. Basically, you're the performer, so you get to choose.
EDIT: I think Biggles answered while I was working on my "dissertation."
