I have a committed partner who lives with me the majority of the time, but I'd still list myself as single. And not bat an eye. We both see other people and describe each other as significant others. Perhaps it would be different if we were married, but we're not.
Unmarried people who have a significant other who agrees with them dating other people is single. Single people go on dates. Multiple dates with other people.
And like london says, there has to be some responsibility of the other person to ask about your situation. Most dating sites aren't the kind where you just read a profile and then meet up. You read a profile...make contact...talk on the site for a week or more...then start texting...then maybe a phone call...THEN meet up. If you can't get to know someone in that amount of time, then you're on your own in my book.
Me - male, 42, poly, straight, in a serious relationship with Audrey, also casually dating.
Audrey - female, 20, poly, pansexual, also casually dating.