Actually, I doubt that this will be a problem. Such language is fairly common, it is known as the most favored nations clause
A lesson that I learned a long time ago, keep your work life and home life separate. The attorneys for your workplace will have the company's interest at heart, not yours. Our company has a fairly liberal computer policy. You can use workplace computers for personal use as long as it does not interfere with work or break any laws. Nonetheless, in our ethics training, it was pointed out that if the company is sued, they may be required to give my computer to the other side. And they will. And anything private on it is open to discovery. They advised keep work and home separate. So I have separate email accounts, separate computers, etc. Never let the two mix.