Language is ambiguous, and we often decide what meaning is correct by relying on context. But sometimes we can rely on grammar.
Let's take your first example:
They criticized my plans to become a musician.
They (plural) cannot become "a musician" (singular), unless by "they" you mean a single person (in a gender-neutral way). Let's presume that from the context it's clear that "they" are several persons. Then it's easy to understand that it is the author of the sentence who wants to become a musician.
Now, the second example:
Researchers criticize plans to change Dutch research organization.
The most likely interpretation is "they criticize plans about changing the research organization." Why? Because we usually complement the word "plans" either with a "to-phrase" or an "of-phrase" or a "for-phrase". The sentence below would look strange:
Researchers criticize plans.
Do they criticize plans in general? This is strange. That's why even the sentence
They criticized my plans to become fat.
Would be understood as "I have plans of becoming fat. They criticized these plans", and not as "They want to become fat. In order to achieve this goal, they criticized my plans".
But again, context is king. We can come up with a context that will change the meaning of the sentence:
I prepared plans for setting up a playground in the park. I presented the plans at a neighborhood meeting. Several members of the meeting wanted to become fat, and viewed my plans as harmful to this goal. They criticized my plans to become fat.