Higher notes revisited: last chance?

Hello!

Let me return to the question I was asking two months ago. It's about
producing high notes and I am not talking here about anything higher than the C3 - the top of the middle register. I don't know how many of you and for how long struggled with the octave key in the beginning but I've been doing this for over three months now.
On the last meeting with my instructor he was puzzled when I was unable to produce notes with the octave key. He checked and said I had a good breath support and he a little corrected my embouchure but I still had general difficulties doing that.
Well, I sometimes manage to 'voice' notes with the octave key but that's a hit and miss process. I am trying my best when practicing but I am not far from where I started four months ago.

Could you please this time comment on the two extracts from:

1. TBA Convention/Clinic 2002 by Rory L.Davis:

'CHECKING PROPER EMBOUCHURE TENSION Have the student finger second space A while the director depresses the octave key. (Don’t allow them to see) If the upper register doesn’t speak or is spread and unfocused, the embouchure is too loose. If only the upper register responds, even upon release, then the embouchure may be too tight, or the instrument may need adjustment.
When the proper embouchure formation and tension are applied, the entire range of the saxophone can be played, with the exception of higher notes, generally E and F. Notes higher than this, as well as palm key notes, will require a higher arch in the tongue similar to an EE syllable. If the lower end of the instrument is unplayable, without embouchure manipulation, then the student have the instrument checked for leaks or alignment issues.'

2. The US Army Field Band Sax Basics by by Sergeant First Class Jeffrey G. Price:

'Octaves and intervals should be changed just with fingerings, not with absurd throat and embouchure movement'.

Both of the above but especially the first extract explicitly suggest to produce
the higher notes (octaves in the second extract, which I admit may be an assumption) with fingering only.

Very often I hear, read and my instructor confirmed that also that to produce
noted with the octave key one needs to 'help' the sax (or oneself) to do that by changing the oral cavity volume or 'voicing' the notes like when singing assuming the embouchure stays the same.

I see a contradiction between 'don't let them see when you press the octave
key and the octave key note should pop up easily' - that's doing nothing other than pressing the octave key and the general advise on 'help the octave key note to be produced by 'voicing' that note' - that's doing something: changing the oral cavity shape, moving Adam's apple up (just a visual affect of changing throat shape I think) like when naturally singing the higher notes, etc. (assuming no change in embouchure of course).

Where is the truth?