Can you please help me understand which is the correct elevation value?

For short-duration activities, you may want to consider setting the watch into Altimeter mode. In that way, all pressure differences detected by the sensor will be assigned to altitude changes, not to weather changes. In short periods of time this assignment probably makes sense, and may therefore provide better altitude information along the activity route considering that, as you stated, the initial altitude is correct.

This comes up a lot and no matter what the answer there will always be exceptions but... the general accuracy sorting (from best to worst) Barometric Altimeter (BA), corrected elevation from data such as DEMS and finally GPS. 

GPS vertical accuracy is often 2x worse than the horizontal (physics) so may well be 10-20m accuracy even with a good fix.  This will not (especially at a 1/sec sample rate) give a total gain with any accuracy. 

With map data, the errors are derived from the initial horizontal GPS error and then the source and quality of the elevation data being used.  DEM data is getting much more accurate, available and better coverage but each platform will use a different source with different quality.  Quality means, among other things, the resolution - for example 5m resolution with 1m vertical accuracy has the data broken into 5m squares (like pixels) and accuracy to 1 m in each square as an average (this is just an example). So this is much better than GPS alone but will not catch all the short ups and downs on a trail for example.  Don't forget that the start point for this is still GPS - so 5m out (a good accuracy for GPS) could still put you over a cliff so the corrected data will be way off.  Also, as noted, you have no idea which source (and quality) the various sites are using. 

Finally the BA has lots of issues as well (mostly related to weather changes and the need to calibrate if you want absolute elevation rather than just change/gain - and what PMO said) but in normal circumstances it is considered the best way to catch all the small ups and downs without any reliance on GPS or an external data source.  You might note that with Garmin Connect, if your watch has a BA, the default is Not to use Elevation Correction and to use it only for those watches without the BA.

So, with your examples, the first two make sense as Garmin will use the same DEMS data based on the same GPS file.  After that, without knowing a lot more about how each app uses the GPX file and how it implements DEMS (and source/quality, cannot explain the results. Some apps will even snap the track to the road/trail and use this location - all the same errors with the addition of assuming the map is accurate.

Probably doesn't really help much but it is what it is.

Thanks for great explanation doesn't Instinct auto-calibrate altimeter after starting activity? Or should I do this manually?

Depends on your settings.  I don't have an instinct (my son does) so I don't spend much time with it specifically.  There are a bundh of calibration settings in the newer watches and, as PMO noted above, a couple of different modes available that primarily determine how the watch handles pressure changes due to weather.  I don't generally calibrate my Fenix prior to or during an activity as I don't really care what the actual elevation is (starting my trail run  at 100m or 200m is not important to me) but only the overall gain.  Things might be different on a long mounting trail or multi day hike - or if I needed elevation data to assist with navigation (not so sure I would want to use my watch for that last one).

Thank you for this detailed reading.

My watch is in altimeter mode as I use it for hiking and I always do a calibration when I arrive to the starting point.

As for DEM accuracy, let's take your cliff example, if the path drawn by my recording is correctly visualized on the map (top of the cliff) can it still measure the bottom of it?

Also I think I've read somewhere that Instinct is constantly checking and matching different values (barmeter, GPS data,etc.) to make autocorrections for the elevation. Is that true? If yes would this require phone connection?

Today I was cycling. Overall it was 56km. Instinct showed 1428m of climbing....in pretty flat terrain. It's not first time when I receive this kind of crazy numbers. After synchronization I used option "elevation correction", suddenly height went down to 385m. Is it possible that wind while riding going into pressure sensor affected overall height? Anyway in many cases Instinct seems to be more toy than real reliable tool. Do you know if it is possible to set anywhere "elevation correction" by default?

The first question is - what was the barometer mode on that activity? Barometer/Altimer/Auto ? 

In MTB Cycling settings I can't find any field related  to pressure sensor.

It's on the barometer settings -> Watch mode