Both discussed paths (tunnel and climbing) take you to the same place: behind a big shelf where you have to jump down. Head north and interact with the trader you meet. Besides standard supplies (various arrows and food) he'll have also very precious Rope Arrows, which you should buy. After the shopping, walk to the Old Quarter Cliffs - Dry Dock climbing on chests.
On the board of The Dawn's Light you can get in two ways. The easier one is to move west and climb where you can see a metal construction. This way let you very quickly pass the Dry Dock and avoid nearby enemies. But you'll also miss a lot of treasures to get.
If you want to explore the Dry Dock, head for the northern hill. Be careful as the path leading on the hill is patrolled by the single guard with a crossbow. Pass him when he stops for a longer moment somewhere. At the very hill you run into another enemy, but he won't be moving. I recommend to rob enemies and take their Coin Purses - 5+9+7+2G, and also search this are for a Sawtooth Arrows and Ashtrays - 8G; you have to drag out another guard from the campfire.
Avoid both enemies and head west. Use the claw. Here, find the Gold Coins - 20+1G and a Coin Purse - 2G. Then stand at the edge. In the place shown on the screen above, you can use the Rope Arrow. Gain momentum, jump to it and climb up. Jump into the small tunnel where you find a Syringe - 10G and a Gilded Bottle - 40G. Now you can jump to the new rope and slide down.
Both paths let you begin exploration of The Dawn's Light - Treadwheel Deck. Choice of the path won't influence the further playing, because you'll get to the same place after all.