In all examples above I have NOT chosen a "part" or what it's called.
I see no logic in choosing "Right1" when I'm not gonna play one side or the other on the keyboard?
Mabe there is a mistake in there somewhere?
The 'mistake' is in your logic!
When you arm a channel to REC, the Part Assignment popup window opens showing the part that will provide the source of the notes to record. It normally opens with the default part highlighted for that channel, and unless you change it, this is where the recorder expects to get the notes from for the recording.
For example the default part for channel 5 is Multi Pad #1, and unless you change it to the part that you actually want to use to provide the notes for that channel recording (say to R1), then when you start recording, the recorder is only 'listening' to the Multi Pad #1 part. No notes from any other part (eg. R1) will be recorded!
You are not selecting the part for playback here, you are simply setting the part that you want to play when you make the recording and therefore will provide the data to record!
So you must set the part in the popup to coincide with the part that you actually want to play when recording the channel. If you intend to use R1 as the source of the notes to be recorded when you record the various channels, you must set the part in the popup to R1 each time! Then the notes that you play using the R1 voice will be recorded.
Unless the default part happens to be the one that you want to use to record the channel, just closing the popup without changing the part (as you say you are doing) will simply select the default part as the source of the recorded data! If there is no output from that part, then nothing will be recorded!
There are instructions about Multi Track MIDI recording on page 69 (English version) of the 'PSR-S970/S770 Owner's Manual', but Yamaha have stupidly chosen to hide away the list of Channel Default Parts in the 'Reference Manual' which has to be downloaded! For convenience I have included them below.
The default recording parts for each channel are:
Channel 1 : Right 1
Channel 2 : Left
Channel 3 : Right 2
Channel 4 : -
Channel 5 : Multi Pad 1
Channel 6 : Multi Pad 2
Channel 7 : Multi Pad 3
Channel 8 : Multi Pad 4
Channel 9 : Rhythm 1
Channel 10 : Rhythm 2
Channel 11 : Bass
Channel 12 : Chord 1
Channel 13 : Chord 2
Channel 14 : Pad
Channel 15 : Phrase 1
Channel 16 - Phrase 2
Regards
Ian