Same number, different setting
A treadmill gives you a moving belt, a pace display, and control over incline. Outside, you choose a route and meet turns, crossings, weather, and changing surfaces. A matching pace number does not mean every part of the two runs was identical. Keep the setting in your training notes.
Compare like with like
If you are following progress on a treadmill, note the machine, incline, and how the session was structured. Include stops or changes instead of hiding them in an average. Outside, keep route and conditions alongside the time. These details help your coach understand the session without expecting a single number to do all the explaining.
Do not force a conversion
There is no universal conversion in this article between a treadmill setting and every outdoor run. Trying to make the numbers match can distract from the actual assignment. Use the intensity and structure your coach planned for the environment you have. Follow the machine's safety instructions and learn its controls before starting.
A better note than fast or slow
Try a description such as steady treadmill run, familiar machine, planned incline, no stops. Or note an outdoor loop with several road crossings. You are not making excuses for the number. You are giving it the context it needs to be useful.
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