It's a matter of emphasis:
When I say "cuido de mí mismo", I'm using "mismo" as an emphasis, but I could have simply said "cuido de mí" and the sentence would have the same meaning.
The second sentence is fine with "lo que tú dices de ti". You can add "mismo" as an emphasis too.
You can also say "lo que tú te dices a ti mismo", but this one needs forcibly "mismo" because you'd be using a reflexive form of the verb.
The first sentence is slightly tricky because of the construction with a verb like "gustar", where the "priority" is the indirect object rather than the subject. "Me gusto a mí" and "me gusto a mí mismo" may have different meanings:
- "Me gusto a mí" for me, it says that I like myself and I don't have to be liked by anyone else.
- "Me gusto a mí mismo" means to me that I accept myself and feel comfortable in my skin.
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