As pointed out, they are certainly not the same, but it might not be a bad idea to track and report on the direct traffic. If there has been outreach done and the company is mentioned in print with a URL, direct traffic (along with some search traffic on the URL or business name itself) is likely to go up. If your email newsletters are not tagged, they're likely to show up under direct traffic. Depending on your role, some of what you do under the greater SEO/inbound marketing role can show up under the direct traffic.
If we can do that effectively, it is in the best interest of the search engine to serve our content to our target audience for us, providing highly targeted visitors to our pages at no incremental cost. This is the secret to the organic channel, and why organic search listings are so valuable. Other channels often come with additional cost for each new visitor provided.
At our agency, we work with sites of varying sizes, from very large to quite small, and recently, we have noticed a trend at the enterprise level. These sites aren’t relying as much on Google for traffic any more. Not only are they not relying on Google traffic, but also, the sites are getting less than 10 percent (or slightly more) of their organic traffic from the search giant.