Akward pose in Object Mode

You need to reset the transform of your bones in Pose mode.

To do this, select the rig and enter pose mode. You should see it when clicking on "Object Mode" in the top left of the screenshot:

After entering Pose Mode, make sure all bones are selected by hitting A.

Then, go to Pose -> Clear Transform -> All

After clearing the transform on the bones, they will appear in the same way as they did in Edit mode.