The example sentences list the following as an example to the former: 私は自分の名前が好きだ。However, I was wondering what the difference (despite some repetitiveness) would be by writing 私は私の名前が好きだ。instead.
Aside from the awkward repetitiveness, I wonder if the nuance is a bit different, such as “I like my own name” vs “I like my name.”
自分 has a lot of functions (which you can read about here) and isn’t really a straight personal pronoun the way that 私 is. The nuances and functions will become clear over time if you read/listen to Japanese.
I like my own name
I like my name
Note that these translations are kind of forced as to show the nuance. It isn’t really possible to show the nuances of Japanese pronouns in English (吾輩は猫である being translated as “I Am a Cat” probably being the most famous example of this problem). As such, there are multiple Japanese sentences that would correspond to the English sentence “I like my name”.
Another issue here as that we have absolutely no context for translating so there is always the looming question of why “私は” is said in both sentences. The answer in this case being because it is pedagogically designed for English speaking novice learners of Japanese but assuming it were a “real sentence” then, again, it changes things.
Since the nuance here is hard to explain without making it sound more complicated than it is this is probably safe to put under the umbrella of things that are better learnt through experience than trying to fully commit to memory the first time you encounter it.
I hope that somewhat helped.