How can I make one button a different color from the other buttons on my site?

You probably need to add more specificity or more accurate targeting to override the existing block button styles. In the first case you are applying the styles to the .wp-block-button__link subelement, but not doing this for the orange buttons. Without seeing the page, I’d say you would need something like this (if the custom class is applied to the block button element):

.wp-block-button.alert-button .wp-block-button__link {
    background-color: #ffa500;
    color: #ffa500;
}

or perhaps (if the custom class is applied to a subelement):

.wp-block-button .alert-button .wp-block-button__link {
    background-color: #ffa500;
    color: #ffa500;
}