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Bob King wrote a nice article on this two years ago for Sky & Telescope:
https://www.skyandtelescope.com/astronomy-news/observing-news/thinnest-crescent-moon/
Short answer: you can usually see the Moon at an age of 2 days, but it's tough. 

You wrote:
"For instance on 26-1-2020 at 51°N 04°E a program gives Phase=0,3  Age=2.
I looked outside on the given time, direction and height and, no moon visible."

A little typo there: the phase was 0.03 (3% illuminated or 0,03 with the continental decimal separator). While historically, and in challenges (like the ones described in the article above), the age of the Moon in days is considered the key parameter, what counts more fundamentally is the Moon's elongation from the Sun which is equivalent to the Sun-Moon lunar distance, LD. The phase of the Moon is then given by (1-cos(LD))/2. A phase of 1% corresponding to an elongation of 11.5° is a lower limit under excellent conditions. 

Checking the geometry last night in Stellarium for your location, you can see that the Moon was well south of the ecliptic, and the ecliptic was inclined at a rather shallow angle to the horizon. This means that the Moon was very low in the sky even at the start of twilight. With binoculars you might have been able to pick it up, but it would not have been easy. Try again on February 25. Even better on March 26.

Frank Reed