Furuncle or Carbuncle: Given the description of a growing sac that popped with blood then pus, and the location under the arm, a furuncle (boil) or carbuncle is highly likely. These are skin infections typically caused by Staphylococcus aureus, which can be treated with antibiotics like amoxicillin. The fact that it resolved with antibiotics and then recurred in a different location supports this diagnosis.

Folliculitis: An inflammation of the hair follicles, which can lead to pus-filled bumps and could be related to the initial presentation on the head. It's common in areas with high friction or sweat, like under the arm.
Hidradenitis Suppurativa: A chronic skin condition characterized by recurring, painful abscesses in areas with apocrine sweat glands, such as the armpits. The pain when raising the arm could be indicative of this condition, especially if there's a history of similar episodes.
Cutaneous Abscess: A localized collection of pus in the skin, which could be caused by bacteria. The symptoms of a growing sac that eventually ruptures with pus are consistent with an abscess.

Methicillin-Resistant Staphylococcus aureus (MRSA) Infection: Although less likely, MRSA infections can present similarly to other bacterial skin infections but require different treatment. Given the recurrence and the fact that it's painful, especially with movement, ensuring it's not MRSA is crucial.
Tuberculosis (TB) of the Skin: Though rare, TB can cause skin lesions that might resemble abscesses or boils. Given the patient's location in the Philippines, where TB is more prevalent, and the fact that the initial diagnosis was unclear, considering TB is important.
Actinomycosis: A rare bacterial infection that can cause abscesses, especially in the face and neck but can occur elsewhere. It's often associated with poor dental hygiene or oral procedures but can be considered in recurrent or atypical cases.

Erysipelas: A type of skin infection that involves the upper dermis, typically caused by Streptococcus bacteria. It presents as a well-demarcated, red, and painful area but is less likely given the description of a "growing sac" and pus.
Myiasis: Infestation of the skin by fly larvae, which can cause lesions that might resemble boils or abscesses. This is rare and usually associated with exposure to flies in certain environments, but it could be considered in a tropical setting like the Philippines.