Connecting a VGA monitor to new Mac Mini?

I have a perfectly good LCD monitor that has a VGA connector on its input cable. I'm currently using it with an old iMac G4 (Gumdrop) via a mini-display to VGA adapter cable.

I'm replacing the old iMac with a new Mac Mini, which has only Thunderbolt and HDMI connectors available. The Mac Mini comes with an "HDMI to DVI adapter" (Apple's description), but I can't find any other information about that adapter on the Mac Mini description pages, nor can I find any photos of the adapter. I assume that the DVI side of the adapter is a standard DVI (F) female connector, which will accept a standard DVI (M) male connector.

Unless I'm missing something (definitely possible), it would seem that the most direct method of connecting the monitor would be to use a DVI (M) to VGA (F) adapter; plug the DVI side into the Apple-supplied HDMI to DVI adapter, and then use the VGA side of the adapter to connect the monitor.

I've read that some folks have had difficulty connecting VGA monitors to the Mac Mini — something about the HDMI signal not accurately translating to VGA with passive adapters. I don't know enough about it to say whether it's workable, but something doesn't add up. If Apple's HDMI to DVI adapter works then it should just provide DVI at the DVI end of the adapter. And if DVI converts to VGA (which it does...I've done it many times), how can the [HDMI to DVI] + [DVI to VGA] path not work?

Are there "different kinds" of DVI signals — namely, the kind that can be converted to VGA, and another kind that can't be converted to VGA?

Thanks in advance.