Ingrown Toenail: The location of the pain (distal interphalangeal joint and medial aspect of the nail) and the exacerbation by heat and standing barefoot are classic symptoms for an ingrown toenail. The aching in the joint and around the nail further supports this diagnosis, as ingrown toenails can cause localized inflammation and pain.

Gout: Although less common in the toes other than the big toe, gout can affect any joint and is known for causing severe pain, especially with weight-bearing activities like standing. Heat can exacerbate gout symptoms, making this a plausible diagnosis.
Osteoarthritis: This condition can affect any joint, including the distal interphalangeal joints of the toes. Pain that worsens with standing (especially barefoot, which may increase pressure on the joint) and aching in the joint are consistent with osteoarthritis.
Trauma or Sprain: Direct injury to the toe could cause pain in the distal interphalangeal joint, especially if the injury involves the joint or surrounding tissues. The pain from trauma could worsen with standing and might be exacerbated by heat due to increased inflammation.

Osteomyelitis or Septic Arthritis: Although less common, infections of the bone (osteomyelitis) or joint (septic arthritis) are medical emergencies that require prompt treatment. Symptoms can include pain, redness, warmth, and swelling, which could be exacerbated by heat and weight-bearing activities. Missing these diagnoses could lead to severe consequences, including joint destruction or systemic infection.
Diabetic Neuropathy or Ulcer: In patients with diabetes, neuropathy can cause pain or numbness in the feet, and ulcers can develop, especially over pressure points like the distal interphalangeal joints. These conditions are critical to identify due to the risk of infection and the need for proper wound care.

Psoriatic Arthritis: This autoimmune condition can cause joint pain and inflammation, including in the toes. While less common, it's a consideration, especially if there's a history of psoriasis or other autoimmune diseases.
Reiter's Syndrome (Reactive Arthritis): A form of arthritis that occurs as a reaction to an infection elsewhere in the body, it can cause joint pain, among other symptoms. It's less common but should be considered in the differential diagnosis, especially if there's a recent history of infection.
Tuberculosis of the Bone or Joint: Although rare in many parts of the world, tuberculosis can affect bones and joints, causing chronic pain and other symptoms. This diagnosis would be more likely in individuals with a history of exposure to TB or in endemic areas.