Show as "Status"

I’ve been logging in pretty much daily for the last month and I always leave my login done at my workplace to save time (please don’t say to just log out at work, I have strict security policies and I always leave my workstation in a locked state when I’m absent).

However, this results in my account being shown as Online constantly since the site must be refreshing status.

Would it possible to add a sort of Presence or Status button to show profile as “Online” or “Offline”?
That way, when leaving my workplace, I can set it as “Offline”.

NOTE: While typing this, a concern came up. I’m not sure how 2 logged in sessions will interact with each other. Will the “Online” status override the “Offline” status?

Would being able to set “working hours” for your account be a suitable substitute? I.e. when it isn’t your account’s “working hours” your advertisements get hidden?

I’m afraid not, since I’d like my ads to be online permanently.
What I want to avoid, are users thinking I’m online when I’m not though. They’re sending messages, thinking I’m online and I’m not replying to them.

You could provide “hours of operation” in your trade terms.