Normale Version: Include bb2 vs ab2 vs ab3
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I've seen there are the same include files with different extensions.
What't the difference of those and wh n should I use which one?

There should be only .ab3 extensions. If we forgot some .ab2 or .bb2, ignore them. Use .ab3, it's the newest.
OK, so you're saying that in AmiBlitz3 there should be only .ab3, in AmiBlitz2 only .ab2 and in Blitz2.1 only .bb2 extensions.
And they are neither downwards nor upwards compatible?

I think the AmiBlitz2 includes still had the .bb2 extension, but in general you should use the ones for the version of Blitz you're using. Later includes would be better, but many use features of AmiBlitz3 that aren't available in Blitz 2.1 so you'll probably have trouble compiling them under earlier versions.
Blitz2 -> .bb2 and .txt
Amiblitz2 -> .bb2 and .ab2 and .txt
Amiblitz3 -> .bb2 and .ab2 and .ab3 and .txt

by load and save .txt you loose all informations for the compiler

You should use .ab3 in Amiblitz3 but it can also read and write all other Formats.
Thats all
For Blitz2 and Amiblitz2, only .bb2 could be used for includes, since there was no on-the-fly detokenisation.
Starting with Amiblitz3, you can include .bb2, .ab2 and .ab3.
The includes shipped with the distribution should be .ab3, as this is the preferable format for Amiblitz3. Advantages are that .ab3 is pure ASCII and contains human readable XTRA information about the compiler.
.bb2 is tokenized and does not work with a regular Text Editor or Version Control system.
.ab2 is ASCII but has a separate XTRA file that is binary. The XTRA file format is somewhat mystic and hard to maintain/extend, so we defined .ab3 where everything is one file and text only.
OK, so I can use .bb2 in an AmiBlitz3 environment.
And If I save my code as .bb2 as well I could open and compile this all in BlitzBasic2?

Yes, you can load all formats into Amiblitz3/PED and export them as .bb2 or .asc, which you can load into BlitzBasic2.
However, if the code uses Amiblitz3 features, it will not compile under BlitzBasic2.