Can a Secondary Definition Violate/Negate the First Definition

Yes. There's even a term for the situation where one definition may contradict another, the less prototypical definition being termed a stipulative definition. Wikipedia has good articles on this, though I'll reformat below:

• Dictionary definition / Lexical definition

The lexical definition of a term, also known as the dictionary definition, is the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictionary [and usually listed first or not far from first where there are different senses, in those dictionaries listing in order of frequency of usage].

[Wikipedia]

• Precising definition

A precising definition is a definition that contracts or reduces the scope of the lexical definition of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition.

For example, a dictionary may define the term "student" as

1. anyone attending an educational institution of any type, or
2. anyone who studies something.

However, a movie theater may propose a precising definition for the word "student" of

[3]. any person under the age of 18 enrolled in a local [Borough of Stockport] school

in order to determine who is eligible to receive discounted tickets.

Precising definitions are generally used in contexts where vagueness is unacceptable; many legal definitions are precising definitions, as are company policies. This type of definition is useful in preventing disputes that arise from the involved parties using different definitions of the term in question.

A precising definition is intended to make a vague word more precise so that the word's meaning is not left to the interpretation of the reader or listener. Here is an example:

From a class syllabus:

"Class participation" means attending class, listening attentively, answering and asking questions, and participating in class discussions.

• Stipulative definition

A stipulative definition is similar to a precising definition, but differs in that a stipulative definition may contradict the lexical definition, while a precising definition does not.

[Wikipedia]

So a precising definition tightens up the basic dictionary definition, and obviously some elements accepted by the broader definition will be rejected by the narrower definition. A layman's hurricane may not be a meteorologist's. (And note that those who claim 'this definition is the only one acceptable' are hyper-prescriptivist.)

A stipulative definition may (I'm not sure why Wikipedia doesn't say does) even include elements rejected by the basic dictionary definition ... and certainly by rival stipulative definitions ('sentence', 'phrasal verb' and 'acronym' come to mind, though I'm fairly sure dictionaries aren't all in agreement even on the 'basic' meanings).

While pragmatics requires that we make an effort to adapt to the register in use in the situation obtaining ('similar' means 'of exactly the same shape though not necessarily size' in maths classes, but 'alike in some significant respects' in the real world), often the greatest error is failing to define terms before trying to discuss a topic. 'The dictionary says ...' is the root of much evil.