Two immediate observations, the mountain on the left marked Mt. Washington is actually a mountain called Three Fingered Jack. Mt. Washington is not visible as it is behind the closer North Sister mountain. You can verify this by comparing angles on google earth or your peak finder site. Second the base of Mount Adams is not visible, you are seeing mountains and hills that are quite a bit closer.

Pretty much only the snow covered peak of Mt. Adams is visible in this image. I have software for calculating radio communication links and it shows that this is viable. Keep in mind that at both ends you are just under 2 miles and just over 2 miles elevation on Mount Adams. If you simply combine the heights of both mountains which is approximately 20000 feet you can then calculate the distance to the horizon which is 173 miles.

If you were standing on a Mountain that was 20000 feet tall you could see 173 miles.

The picture was taken on Mount Bachelor, from Mount Bachelor if the rest of the world was at sea level you would be able to see 116 miles. Since Mount Adams also sticks upwards of 11000 feet you can see it.

This is why you can see Mt. Adams as you should be calculating the combined heights of the two mountains. The distance in the image is 154 miles. You can calculate the horizon here http://www.ringbell.co.uk/info/hdist.htm or at the site you linked.

Based on your calculations you should be able to see the top 4300 feet of Mt Adams.

Hope this makes sense.