This is actually a very complicated question, about (1) the scope of "knowingly", (2) what kind of "intent" is required for conviction, (3) how does the jury understand and evaluate concepts of intent and (4) how does an attorney persuade the jury that the situation does or does not satisfy the particular intent requirement. One thing we can dispose of quickly is the possibility that the law says that the accused has to actually know that the act is against the law. It is a standard legal fiction (2,400 years old) that the accused knows the law, or should have known.
As for the 4th element, persuasion, on the strong side we have statements of intent by the accused – "And I pulled out my rifle and blew his head off, and I laughed the whole time". What the jury has to decide is whether the accused had in mind a conscious purpose of doing some act (that is, it wasn't just an accident like butt-dialing, or an instant stimulus-response reaction to some event). "Intent" generally falls in the realm of acting "purposely", which the Model Penal Code §2.02 defines as:
A person acts purposely with respect to a material element of an
offense when:
(i) if the element involves the nature of his conduct or a result
thereof, it is his conscious object to engage in conduct of that
nature or to cause such a result; and
(ii) if the element involves the attendant circumstances, he is aware
of the existence of such circumstances or he believes or hopes that
they exist.
which more or less means what you think "with intent" means. That doesn't say what they intended to do, it just distinguishes intent from negligence and accident. The prosecutor would then present factual evidence that the accused had a bad intent, like showing that he actually made an attempt to access credit card account data (that such data had been accessed when he broke into the system), etc.
We can sort of dispose of the other scope question about "knowingly", namely, what things would he have to know? The chunk accessing a protected computer without authorization can be interpreted in a number of ways, having to do with which elements of the clause are known to the accused. You might know that you were accessing but not know that you were unauthorized; you might know that you were accessing and unauthorized, but not know that the computer is protected. The only reliable way to know which is which is to study the case law on a statute and see if there is a controlling decision that say e.g. that you have to know that you are accessing and are unauthorized, but you don't have to know that the computer is protected. I haven't determined (yet) is there is decisive case law on this, but I'm betting that the outcome would be that not knowing of the protected status of a computer carries no weight.
As for what kind of intent, there is a distinction between "specific intent" and "general intent". The distinction comes down to having some evil purpose like "make him suffer" (general intent) versus a specific evil purpose like "kill him".
Finally, the people who actually decide, the jury, will be given instructions that say what they have to look for. Here is the tip of the tip of the iceberg, from California's criminal jury instructions. The judge will say...
The People must prove not only that the defendant did the acts
charged, but also that (he/she) acted with a particular (intent/
[and/or] mental state). The instruction for (the/each) crime [and
allegation] explains the (intent/ [and/or] mental state) required.
A[n] (intent/ [and/or] mental state) may be proved by circumstantial
evidence.
Before you may rely on circumstantial evidence to conclude that a fact
necessary to find the defendant guilty has been proved, you must be
convinced that the People have proved each fact essential to that
conclusion beyond a reasonable doubt.
Also, before you may rely on circumstantial evidence to conclude that
the defendant had the required (intent/ [and/or] mental state), you
must be convinced that the only reasonable conclusion supported by the
circumstantial evidence is that the defendant had the required
(intent/ [and/or] mental state). If you can draw two or more
reasonable conclusions from the circumstantial evidence, and one of
those reasonable conclusions supports a finding that the defendant did
have the required (intent/ [and/or] mental state) and another
reasonable conclusion supports a finding that the defendant did not,
you must conclude that the required (intent/ [and/or] mental state)
was not proved by the circumstantial evidence. However, when
considering circumstantial evidence, you must accept only reasonable
conclusions and reject any that are unreasonable.
and then there will be some specific elaboration of whether they have to find that the accused just generally intended to do bad, versus intended to specifically defraud.
