Most state laws in the US are based on the Uniform Motor Vehicle code. Usually there will be specific language that states if you are a cyclist on the road, you obey the laws that apply to vehicles, and if you are a cyclist on "pedestrian facilities" eg. sidewalks, sidepaths or multi-use trails, you obey the laws as they apply to pedestrians.

There is also usually phrasing that you must obey "traffic control devices".

In this case if you are on the path, crossing a roadway and continuing on the path, from a legal standpoint you most likely have to obey the signal.

From a practical standpoint, ask yourself "As a pedestrian, what would I do here?", since the pedestrian rules are what most likely apply.

Most state laws in the US are based on the Uniform Motor Vehicle code. Usually there will be specific language that states if you are a cyclist on the road, you obey the laws that apply to vehicles, and if you are a cyclist on "pedestrian facilities" eg. sidewalks, sidepaths or multi-use trails, you obey the laws as they apply to pedestrians.

There is also usually phrasing that you must obey "traffic control devices".

In this case if you are on the path, crossing a roadway and continuing on the path, from a legal standpoint you most likely have to obey the signal.

From a practical standpoint, ask yourself "As a pedestrian, what would I do here?", since the pedestrian rules are what most likely apply.

Edit after O.P. added the picture:

If I am not in the road traveling like a vehicle before the intersection, I would absolutely behave like a pedestrian here - especially if there are similar trees on this side. Any car turning right is going to be barely aware of pedestrians and won't be looking for bikes at all. An on coming car turning left has the same problem, but at least you have a chance to see and avoid them. Their awareness will be even lower if the crosswalk sign indicates "Don't Walk".

If you were to ride out at even 10 MPH you will catch most drivers completely by surprise. This is one of the most common types of collisions.

Most state laws in the US are based on the Uniform Motor Vehicle code. Usually there will be specific language that states if you are a cyclist on the road, you obey the laws that apply to vehicles, and if you are a cyclist on "pedestrian facilities" eg. sidewalks, sidepaths or multi-use trails, you obey the laws as they apply to pedestrians.

In this case if you are on the path, crossing a roadway and continuing on the path, from a legal standpoint you most likely have to obey the signal.

Most state laws in the US are based on the Uniform Motor Vehicle code. Usually there will be specific language that states if you are a cyclist on the road, you obey the laws that apply to vehicles, and if you are a cyclist on "pedestrian facilities" eg. sidewalks, sidepaths or multi-use trails, you obey the laws as they apply to pedestrians.

There is also usually phrasing that you must obey "traffic control devices".

In this case if you are on the path, crossing a roadway and continuing on the path, from a legal standpoint you most likely have to obey the signal.

From a practical standpoint, ask yourself "As a pedestrian, what would I do here?", since the pedestrian rules are what most likely apply.