Does balkanized government code imply in all cases a series of different, separate governments?

Yes. Here's what Book 3 had to say: "Balkanization is a special result, and indicates that there is no world government; instead several rival territorial governments exist. In such cases, the referee should generate the specific qualities of each territory on the planet separately."

Then, in a decade or five of picking up the pieces, they kind of realize that despite their differences, they need to have a body to work on international issues, to prevent large scale conflicts and eventually it adds some aspects of world government to it.

That said, the countries are not merged, they just all contribute members to a ruling council. The council's will carries the days (majority rules, 2/3rds in some particular areas) but there may be some limited 'Notwithstanding' clauses. But they work like the EU Parliament. There are plenty of national politicians in each of the polities on the world that contribute members to the ruling council.

So what type of government is that?

Balkanized?

If the world has a Government type of "7" then this is "Balkanization" government.

Charismatic Oligarchy or Non-Charismatic Oligarchy?
I don't feel like it is an Impersonal Bureaucracy.

Not sure how to code such a world government (with individual nations contributing to it and having their own governments beyond that in most matters).

A bit of "cart before the horse here" but as previous discussions have suggested, this might be a variety of Government types. It could be "Self-Perpetuating Oligarchy" or "Representative Democracy" or "Civil Service Bureaucracy" or "Impersonal Bureaucracy" or "Charismatic Oligarchy" or even perhaps, depending upon the characteristics of the constituent / member "rival governments" and how they choose their representatives on the ruling council, "Feudal Technocracy."

What you really seem to be asking about is the nature of a "confederation," about the relative authority / power of a central government with respect to its "(con)federal" components. The EU, Canada, the U.S. and the Russian Federation might all be classified as "confederations." If you're trying to get a driver's licence in the EU, it looks like a more "unitary" government than does Canada or the U.S. If you're trying to sell alcohol or transport hydrocarbons across one of the "internal boundaries" then the EU and the U.S. seem more "unitary" than does Canada. If you're trying to build a capital plan for the construction of a hospital or elect a head-of-government then perhaps Canada seems to be the most "unitary" government as compared to the EU or the U.S. If you're trying to fight a war, then Canada and the U.S. seem more "unitary" than does the EU. If you're a "foreigner" trying to get a visa, then it's difficult to tell much difference between Canada, the U.S. and the EU.