There is no good rice or bad rice nor is there any healthy or unhealthy food, this is a very old and rather vague concept. If a person is focused on good health then a good and balanced diet should be vouched for instead. If something is edible and has no toxic effect then it is food, the nutritional requirements chart after that needs to be analyzed according to one's physical needs and planned. You can consume 'junk food' as well but that requires you to balance your overall diet by making up for that extra fat you consumed, that is if you are actually focused on good health.
Like you said, if White Rice has more Carbohydrates while other Rice types have less Carbohydrates and more proteins then choose any but balance the diet accordingly. So for example if you choose White Rice then make up for the lost protein by eating foods containing protein, likewise if you choose wild rice then make up for the lost carbohydrates by eating other foods. And of course then there are other Nutrients as well.
The entire objective is to obtain nutrients, however way you do.