What could pain in the foot running along the second from the outer metatarsal in the foot mean, when it is weight bearing?

Metatarsalgia: That is a big fancy word for pain in the ball of the foot, metatarsal(s) specifically. It can arise from overuse, worn out/ faulty shoes, foot structure etc. Look at your shoes (to see if they need replacement or an arch support) and try an over-the-counter antiinflammatory. If these do not help, consider seeing a podiatrist.

Metatarsal pain: Pain in the ball of your foot is most often mechanical, meaning the way your foot hits the ground. Your third metatarsal, as you have described it, may become inflamed due to pronation. The result can be a painful joint, stress fracture, bursitis, tendonitis, and possibly even a neuroma (pinched nerve). You should see a podiatrist for treatment.