Well, let's go step by step. In your first sentence, you state the phrase that you want to translate as "I want to do it by myself." Now, I may be wrong, but I think there's a slight difference between saying that and "I want to do it myself." Both have a sense that you want to do it yourself, but when you add "by" the emphasis goes to the fact that you don't want any help, that you want to do it alone. The second variation, "I want to do it myself" puts, according to me, more emphasis on the fact that you want to do it with your own hands, but some help can and may be welcome. So, for these two first options, I would say: "I want to do it by myself." = "Quiero hacerlo yo solo." or "Quiero hacerlo por mí mismo, pero solo." "I want to do it myself." = "Quiero hacerlo yo mismo." or "Quiero hacerlo por mí mismo." Now, let's get on to the correction you got from a native speaker, "Quiero hacerlo para mí mismo." The expression, as you know, does exist, but I think the native speaker was wrong or misunderstood you (and I say this respectfully). If you want to say "I want to do it by myself", I would definitely not use "Quiero hacerlo para mí mismo." Having said this, the situation where you would use "para" would be something like this: "Why are you baking cookies?" and you would answer "Oh, I'm baking them just for myself." Then you would say: "Por qué estás horneando galletas?" "Ah, solamente las estoy horneando para mí (mismo)." The expression "para mí mismo" can be used in other situations, that are similar to the one I just wrote. Here are some examples: "Ella cantaba para sí misma por las mañanas." = "She sang to herself in the mornings." "Dijo las palabras solamente para sí mismo." = "He uttered the words just to himself."

Es incorrecto decir "Quiero hacerlo para mí mismo" y también incorrecto "Quiere tratar de hacerlo por su mismo", es por sí mismo, por ti mismo, mí mismo...

Quiero hacerlo por mí mismo = Quiero hacerlo yo solo, sin ayuda de nadie más. Por ejemplo, reparar el coche. Quiero hacerlo para mí = Quiero hacerlo para mi propio beneficio, para mi bienestar personal. Por ejemplo, practicar yoga todos los días.