DIAGNOSIS: You're right to be cautious. The screen has a special anti-glare and anti-reflection coating that can be damaged by chemicals or pressure.
FIX: Use only a dry, soft microfiber cloth first, like the kind for glasses or camera lenses. Gently wipe in one direction. If you need moisture, dampen a second microfiber cloth with distilled water only—tap water has minerals that can leave streaks or damage the coating. Wring it out completely so it's barely damp, never wet. Wipe gently, then immediately dry with the dry cloth.
IF THAT DIDN'T WORK: For stubborn spots, use a 50/50 mix of distilled white vinegar and distilled water on your damp cloth. Never spray anything directly onto the screen; always apply to the cloth. Avoid circular motions.
HEADS UP: Never use window cleaners, alcohol, ammonia, or any abrasive cloths (paper towels, tissues). They will permanently haze or strip the coating. Also, don't press hard near the edges of the screen, as that's where the panel is most vulnerable to pressure marks.