Obese is just a medical term. BMI is a height to weight ratio so it’s accurate in that way. It does miss as a measure of over-fat vs. overweight, but more often in the other direction (fat people with normal BMI) but it can err in the other direction sometimes, sure.

The easiest and most accurate way to tell if you are over-fat is waist to height ratio. Your waist circumference should be less than half your height. So if you are 180cm, waist needs to be less than 90cm. This is the only stat that’s been a reliable predictor for health problems from too much fat (because it’s the torso fat that’s more problematic health-wise)

The other thing to remember is that the risk curve for underweight is steep but for overweight it’s shallow - a couple extra kg has almost no extra health risk.