B737 Captain reported inadvertent escape slide deployment during boarding at the gate.

Narrative

While parked at Gate XX in ZZZ; we closed the main cabin door and were preparing to secure the cockpit door when a passenger approached the Lead Flight Attendant. The passenger stated they had checked their bag at the gate; not realizing it would be sent through to their final destination. They informed the Lead that the bag contained essential medication and special dietary items required every three hours.As this was the final bag to be sent out; and the cabin door had just been closed; I contacted ground personnel to see if the bag could be tagged for retrieval in ZZZ1. During a brief discussion; the Lead Flight Attendant advised they could make space in the cabin for the bag. I then asked ground if the bag could be retrieved; and they responded affirmatively; immediately calling for the jet bridge to return to the aircraft.Within moments (much quicker than expected) and as the bridge approached; the sun had just risen and was shining horizontally through the cockpit and cabin windows; likely impairing visibility of the door status from the exterior by the gate agent. I observed the flight attendants rearranging some carry ons to accommodate the passenger's bag. Shortly afterward; I heard the familiar sound of the cabin door opening; immediately followed by some commotion. Upon stepping out of the cockpit; I discovered the main cabin door partially open; with the emergency slide lying on the floor of the jet bridge. The red Armed" warning ribbon was still visible across the door window. The gate agent was physically holding the door; which was under some pressure to open further. I immediately instructed the flight attendants to clear the forward galley and contacted Maintenance to report the situation and request guidance. We agreed that the agent should clear the area for safety. I returned to the galley and held the door steady from the inside; assessing that the primary risk was to personnel outside the aircraft. Once the agent was clear; I slowly released the door; which moved very little. Importantly; the slide did not inflate.I made a PA to inform passengers of the situation and requested their patience as we evaluated our next steps. My First Officer and I continued coordination efforts with Maintenance; Operations; the cabin crew; ground staff; and passengers. Given that Door L1 was no longer safe to use; we requested air stairs to be brought to the rear of the aircraft in case a full deplaning became necessary. Operations responded promptly and prepared to deplane passengers via Gate XY; using stairs; or elevator as needed. I did not consider it safe or appropriate to have passengers re-enter the jet bridge with the slide exposed and unsecured. Maintenance arrived shortly thereafter and assessed the slide. The technician quickly determined there was minimal risk of deployment. According to him; the slide had been "packed incorrectly." He successfully removed the entire slide assembly; confirming the safety pin was still secure; and stowed it in operations for return to a maintenance base. We proceeded with the rear deplaning without incident.Notably; the gate agent who opened the door handled the situation with exceptional composure and professionalism. He demonstrated a strong commitment to customer service and worked diligently to assist our guests throughout the event. There may have been some pressure on behalf of the agent to make sure they were not the cause of any delay. That said; we still had 10 minutes to open and close the door before our scheduled departure time. I felt we had more than sufficient time to successfully open and close the door and still get an on time if not early departure."