I want to go with translation tables as described here in the third example.
This is not hard to implement but what I wonder is basically how do I want to encode those languages? I looked up the ISO 639-3 files that contain the latest languages and their codes but is it a good idea to include them all?
The Language table is supposed to provide all kinds of languages for Stores. Those stores are allowed to decide themselves which languages they want to support. However, my database is going to tell them how many languages there are.
So, is there a "common used" list of languages? I don't think that Facebook and/or Google really support 7866 languages which is the number of languages listed in ISO 639-3.
Or would I use the Language Culture Names codes like en-UK, en-US, de-AT, etc.?
If I understand your question, you're asking if you should make a reference table with 7866 rows, one for each language in ISO 639-3. I don't really see the downside as it's not going to incur much storage or performance cost, compared to just storing a subset.
The real question is which languages you want to actually translate and whether you want to support dialects. If you don't, you can just use languages like en, fr, etc, and you could just store those in the reference table if you wanted to (though the savings are minimal).
If you want dialects, then you would need en-us, en-gb etc. Since these should fall back to the non-dialect version (ie just en), it would be a good idea for your columns to distinguish between the language and the dialect. So you can store "en | us", "en | gb" (possibly "en | null" too) and it will be easy to provide translations as overrides, but with most of them falling back to a default dialect.