I did some more tests with photos tonight. One thing is that the lighting was quite bright, and reducing the amount of light did improve things a little. However, reds weren't as blown out using my old Canon point and shoot compared to my iPhone 5. For the iPhone, using different camera apps didn't seem to make much difference.
The red Clacks in the following pictures are Revenge Red (more of standard red, a little deeper red than the HHKB red keys) and F2 Red (darker maroon red). The Canon pic looks the most accurate on my monitor, although the reds are still quite off... Revenge Red almost looks like Hack Orange.
iPhone 5 with full lights:
iPhone 5 with less lighting:
Canon point and shoot with less lighting: