You are both correct and incorrect. Itachi gave himself the sickness because of how he trained/prepared himself before each battle. Itachi was a strong fighter and trained obsessively next to his friend shisui, but after losing shisui and gaining the mongekyo, he unlocked the tsukuyomi, his strongest TRAINING TOOL, but it came at a cost. It is said that when you're in the tsukuyomi, you can perceive time differently. Itachi uses Tsukuyomi on Izumi Uchiha, causing her to live through decades of a happy life in the span of a few moments. When she dies of old age within the illusion, her actual body dies as well. Because of this, it is evident that using the illusion you can change your surroundings to anything you want or envision anyone in the illusion you wish. Itachi would use this to his benefit by using a crow clone to cast tsukuyomi on himself to prepare for any situation or using his currently knowledge of any opponent to train against what they might do. However, it is also stated that just simply being in the tsukuyomi places a great strain on your body because of the jutsu itself, regardless of how it's used. So him using the tsukuyomi to train is what gave him the sickness.