Three Different Flavors

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This leads to the question if you have to use DIME to benefit from AGP. The answer even provided by Intel is 'NO'.

You can use AGP without using the DIME feature at all. In this case the graphic accelerator is just benefiting from the much higher transfer rates than PCI. The 'sidebands' can be used, but they don't have to. Without 'sidebanding' the transfer rate is already 266 MB/s, which is double of what a stand alone PCI graphics card would get. Here the access can (as with PCI cards as well) either use PIO or DMA to transfer the data from main memory into the frame buffer of the graphic accelerator.
The majority of graphic accelerators will most likely use DIME, thus saving on board texture memory and hence making the card cheaper without loosing performance. Of course these cards should be using the 'sidebands' to enable 'x2' mode.
The high end versions will most likely use DIMEL (Direct Memory Execute and Local also). Often used textures would be stored in a (large) on board local memory, less frequently used ones would reside in the AGP RAM. These cards will come with a lot of memory on board, like e.g. the (expensive but fast) Diamond Fire GL 4000 (PCI) with its 32 MB RAM already shows. Even Intel admits that high end solutions will still have a very large local memory, but will be too expensive for mainstream.