Can anybody help determining it from the photos please?
Sorry but it's impossible to identify finishes from appearance only. It can be hard to ID them in the flesh!
One can make guesses1 based on various clues (including the age of something or the date of the last refinish) but for a thin coloured finish any of around 4-6 options2 are possible and could give very similar results — similar enough that you couldn't tell for sure what had been used in a blind test. Which is why I'm surprised none of the varnishes you tried came even close, but maybe you just didn't have the right colour to hand.
And it is the colour that is the most critical thing to match, not the type of finish.
Finish type doesn't matter
Within reason, the type of finish you use doesn't matter when all you're trying to do is match the appearance. So for example you can retouch old varnished or shellacked pieces with varnish, "gel stain", shellac or lacquer and nobody would know.
Consider a complete refinish
I know renewing the handrails is what you want to do or have been asked to do but consider refinishing the whole balustrade.
This was not finished to a high standard initially anyway (I hope it was a DIY job and no professional was responsible for it!), and the surface is a bit tired and worn all over, with dings, scrapes and what looks like some flaking, so all of it could do with having new finish applied. And of course this option neatly sidesteps the issue of colour matching :-)
If this is out of the question, then....
Deliberately don't try to match the colour
There are few things that look worse than a poor attempt at a colour match on a portion of something, with the rest right there to highlight that the colour is off somehow3.
So instead deliberately make the handrails darker or lighter. There is already much precedent for this as a decorating theme, with handrails sometimes stained much darker (possibly initially to help hide wear or grime, and then it became a style), or painted darker (up to black) or lighter (all the way to white):
1 You can investigate further by trying solubility tests (one of the few ways one can distinguish some finishes from others) but as outlined above, this isn't actually necessary so I don't think there's any point.
2 Not a comprehensive list: pigmented oil-based varnish, oil-based varnish coloured with dye, pigmented waterbased varnish, waterbased varnish coloured with dye, "gel stain".
3 And there are multiple ways to be off, slightly too yellow/orange, slightly too red, slightly too brown, just the right hue but too dark, or a combination.