I have a similar (or the same) model and that second accuracy is super useful for timing coffee runs and pee breaks in between calls. Absolutely love it. Best watch Iâ(TM)ve ever owned. Someone above commented how it does its one job flawlessly and they are absolutely right. The date always being accurate and accounting for the month (even in leap years) is very useful as well.

Unless you're hosting a website from behind that router and blocking twitter.com in the referrer header, OR an IP that twitter.com resolves to is trying to open connections to your router, your router has nothing to do with traffic from Twitter.