Whisk together vinegar, 2 tablespoons lemon juice, 2 teaspoons mustard, and garlic in a bowl. Slowly drizzle in 3 tablespoons olive oil, whisking constantly, until mixture is slightly thickened. Add salt and pepper to taste. Add red onion, tossing to coat. Let stand 30 minutes (can be made ahead and refrigerated).
Combine cream cheese and next 4 ingredients. Stir in 1/4 teaspoon pepper, remaining 1 tablespoon lemon juice, and remaining 1 teaspoon mustard.
Slice ciabatta horizontally into top and bottom halves. Spread cream cheese mixture evenly on both pieces of bread. Arrange salmon in an even layer over cheese on bottom half of bread; press salmon firmly into cheese. Top with spinach and marinated red onion. Place top half of bread on sandwich can be prepared in advance up to this point if kept tightly wrapped and refrigerated 4 to 6 hours).
Cut into 4 individual sandwiches. Heat panini maker or grill to medium-high heat. Brush both sides of each sandwich with olive oil. Grill about 5 minutes, or until crisp and browned. (If grilling, weigh down sandwiches with a baking sheet topped with heavy canned goods.) Cut each sandwich in half diagonally.
Note: To serve as hors d'oeuvres, cut each grilled sandwich into 3 or 4 rectangles or "fingers."