The colors "yellow" and "cyan" are impossible to read on the white background.

May I suggest that you use class="ansicolor-cyan" instead of style="color: cyan"? This would allow someone to override the exact saturation/value of a color to make them readable.

By default, I'd suggest the following colors:

.ansicolor-cyan {
color: #0AA;
}

.ansicolor-yellow {
color: #AA0;
}

I don't have an example of all the different colors, magenta may also be a problem...

You could do this with all the various attributes, including 256 color ansi sequences, bold, underline, bright, etc.

.ansicolor-bold {
font-weight: bold;
}

.ansicolor-underline {
text-decoration: underline;
}

.ansicolor-bright {
...hmm...not sure...
}

I'd be happy to help with the CSS...