How high is the fence allowed to be?

You are right, the decisive factor is the State Building Code, here the Bavarian Building Code (BayBo), see Article 9 and Art. 63(6).

Which version is that?

If the OP has quoted everything regarding enclosures from the development plan and there are no other local building regulations (design statutes etc.), the BayBO applies:

Art. 6 Setback areas, distances
...
(9)
Within the setback areas of a building as well as without own setback areas, even if they are not attached to the property boundary or to the building, the following are permitted
...
3. Retaining walls and closed enclosures in commercial and industrial areas, outside of these building zones with a height of up to 2 m.

and

Art. 57 Building projects exempt from procedure, removal of installations
...
7. the following walls and enclosures:
a) Walls including retaining walls and enclosures, privacy fences and terrace partition walls with a height of up to 2 m, except in open countryside, ...

If you erect the enclosure on your own authority, i.e. without the involvement of the neighbor, which is not recommended, you must build it entirely on your property, you are solely responsible for its construction and maintenance, and you alone bear the costs for its creation. The height counts from the natural ground level of your property to the top edge of the wall/fence. Up to 2 m means including 2 m. But since the ground surface is the reference, we are talking about a tolerance range of approx. 5 cm (in case of dispute, a court might apply a different tolerance).

Fence height is regulated by the State Building Code and not by the Neighborhood Act

It depends on the federal state. In NRW, for example, both must be considered; in Bavaria, there is no Neighborhood Act.

So according to your statement a wall may be 2m high but a fence then not?

Somewhere it must surely say that fences are equated with walls.

No, I did not say that a wall may have a different height than a fence. I said that the specification from c) has nothing to do with case a).

These are completely two different things: a fence is usually treated the same as a wall, but not a RETAINING wall - and only the latter is mentioned in c).

A RETAINING wall is not a boundary wall nor a masonry base of a fence, but supports a stepped slope to prevent sliding. If such terracing of slopes should not be laid out with step heights of more than 2m, this has nothing to do with boundary fences (or walls, or fences with masonry bases). That is why this stuff is also under different letters.

In the present case, c) only limits retaining walls to 2 m height (or beyond that are not allowed without a permit) and designates boundary enclosures (in the interior area) as allowed without a permit and without height restriction.

From the consideration of the special nature of retaining walls (namely: that they must technically be walls) no unequal treatment of boundary enclosures depending on their material is derived.

Which version is that?

Got you, it was the first Google hit, apparently the version from August 4, 1997, not the latest. With linking, some things would be easier here ...

in Bayern gibt es kein Nachbarschaftsgesetz

As far as I know, there is a law in Bavaria for the implementation of the Civil Code, whose seventh section regulates neighbor law.