This article is wrong on several points. “Je quitte la France” and “je pars de France” are often interchangeable, and where they aren't the article is missing some important points.
In the sense of “I'm emigrating from France”, i..e leaving and not coming back, quitter is more common, but I don't think partir is wrong. There are Google Books hits for “quitte la France pour toujours” (this includes both the present quitte and and the past quitté), but a single web hit each for “pars de France pour toujours” and “parti de France pour toujours” and none for “part de France pour toujours”. The phrase “love it or leave it”, which originated in the US, made its way to France around 2007 as “La France, tu l'aimes ou tu la quittes”. But despite the prevalence of quitter, as a native French speaker, I can't find anything wrong with “je pars de France et je ne reviendrai jamais”. I think that quitter is a bit more official or emotional, while partir is a bit more casual, and that makes quitter generally more suitable if you're leaving and not coming back.
In the sense of “I'm leaving France temporarily”, I think quitter and partir are interchangeable. I don't feel any difference between “je quitte la France jusqu'à ce que l'affaire soit prescrite” and “je pars de France jusqu'à ce que l'affaire soit prescrite” (“I'm leaving France until the statute of limitations expires”), or between “je quitte la France pour quelques jours” and “je pars de France pour quelques jours”.
In the sense of “I'm crossing the border”, the correct verb is quitter. In some places where a road crosses a French border, there's a sign “vous quittez la France” (photos: 1, 2). Another possible verb is sortir. It's mostly used in legal contexts and we don't usually say “sortir de France”. A minor needs an autorisation de sortir du territoire.
In contrast, to indicate the point of origin of a trip, the correct verb is partir. For example “Le Tour de France partira de France en 2020, mais du Danemark en 2021” (“the Tour de France [long-distance cycling race] will start from France in 2020, but from Denmark in 2021”). Here quitter is clearly wrong, or more precisely it would have a different, uncommon meaning: “le Tour de France quitte la France” would mean that the entire race is moving to some other country (as in “le Paris-Dakar a quitté l'Europe et l'Afrique pour l'Amérique du Sud en 2009*”).
Partir is also the correct verb in the sense of “setting into motion”. I can't think of a good example with such a large area as a country. For example, “le train part de Paris” refers to the time when the train sets into motion in a station in Paris, whereas “le train quitte Paris”, strictly speaking, refers to the time when the train leaves the Paris city limits. In practice the two might be used interchangeably if Paris is used as a metonymy for that station. A sentence like “le train quitte Paris à 15h24” would indicate the time of departure from the station; it wouldn't be wrong, but “le train part de Paris à 15h24” is more common.