> The article makes an impression that "gezellig" is somewhat close to French "savoir-vivre", "la vie est belle", Italian "dolce vita", and maybe even Latin "carpe diem"?

I don't speak those languages, but I don't think those words are an apt translation. The word doesn't really describe a way of life, or the enjoyment of life in particular, it's more of a vibe or an emotional state or an emotional association to a certain place, event, or situation. The English word "convivial" seems to come pretty close, though from the example sentences I can find I still think the meaning is slightly different.

I find it quite difficult to accurately describe my perception of the word "gezellig" without anecdotes of shared experiences. I believe savoir-vivre and dolce vita are similar in that way; there are direct translations, but they all miss some element of what the original meant.

No, "gezellig" is like just being with a bunch of friends having fun. Everybody is happy talking to each other, mood is great, maybe some food and drinks available.

At least as a native that's how I always interpret it. It's always in a setting with people.