When does a bathtub trap become a running trap?

[smartnewb]

I'm putting in a laundry closet on may main floor. It shares a wall with a bathtub in another room, which drains horizontally through several floor joists before joining the vent stack at a wall 5' away. (See picture.)



As you can see, it's a tight squeeze. I'd like to add drainage for the washer and laundry sink to the same horizontal pipe currently used by the bathtub. As I see it, there are two options.

Option 1. Have all three fixtures (washer, laundry sink, bathtub) drain to the same trap. This would require moving the existing trap 6" to 1' for better accessibility.

That brings me to the title of this post: I know running traps are not to code, and traps are supposed to be located directly below a drain. But modern bathtub drains run horizontally for a few inches before they drop into a trap. (See the horizontal section of pipe circled in red below.)



Surely, these bathtub drains are legal -- every single house in North America has one! So the question is: how much horizontal run is allowed before a trap becomes a "running" trap?

Option 2. Add a second trap (for the washer and laundry sink) to the same horizontal drain pipe. According to what I've read, multiple traps are allowed to connect to the same horizontal drain pipe only if you install an extra circuit vent for each additional fixture. In my case, it would be extremely difficult to install an extra circuit vent. Can I get away without one?

Thanks for reading. I'm a long-time listener and first-time caller. I've learned lots from this site so far. It's a great resource! (Notwithstanding the troubles I had uploading images...)