Acute Appendicitis: The presence of tenderness in the right lower quadrant, along with symptoms of abdominal pain, vomiting, and a history of loose bowel movements (LBM), points towards appendicitis. The absence of Rovsing's sign and the psoas sign being positive also supports this diagnosis, as these signs can be indicative of an inflamed appendix irritating the surrounding structures.

Gastroenteritis: Given the symptoms of abdominal pain, vomiting, and LBM, gastroenteritis is a plausible diagnosis. The tenderness in the epigastric and hypogastric regions could be due to the inflammation of the stomach and intestines.
Urinary Tract Infection (UTI): The presence of hypogastric tenderness could suggest a UTI, especially if the patient is experiencing dysuria or frequency, which are not mentioned but should be considered.
Mesenteric Lymphadenitis: This condition can mimic appendicitis and is often associated with viral infections. It could explain the abdominal pain and tenderness, especially if the patient has recently been ill.

Ectopic Pregnancy: Although less likely in a 13-year-old, it's crucial not to miss this diagnosis due to its severe consequences. A pregnancy test should be considered, especially if there's a possibility of sexual activity.
Intussusception: A condition where a part of the intestine slides into an adjacent part, which can cause severe abdominal pain and vomiting. It's more common in younger children but can occur in adolescents.
Ovarian Torsion: This is a surgical emergency that can cause severe abdominal pain. The absence of specific signs like the obturator sign does not rule out ovarian torsion, and a high index of suspicion is needed.

Inflammatory Bowel Disease (IBD): Conditions like Crohn's disease or ulcerative colitis can present with chronic abdominal pain, diarrhea, and weight loss. While less likely to present acutely, they should be considered in the differential diagnosis of chronic or recurrent abdominal pain.
Hernias: Although not directly indicated by the symptoms provided, hernias (especially if incarcerated or strangulated) can cause severe abdominal pain and should be considered, especially if there's a history of hernia or specific findings on examination.