The mens rea element of an offence almost never requires that the accused know what they're doing is a crime. And that is why ignorance of the law is no excuse.
As Glanville Williams writes in Criminal Law: The General Part, 2nd ed. (1961), p. 30:
a man may have mens rea although he is ignorant of the law. ... the mens rea of English law does not necessarily connote an intention to engage in moral wrongdoing. ... In this respect the phrase mens rea is somewhat misleading.
And in footnote 3, he observes that the modern view of mens rea is simply "an intention [or recklessness] to do the act which is made penal by statute."
The modern view is that mens rea is simply whatever fault element is required by the law to make out an offence. See Williams, p. 31. Mens rea is:
the mental element necessary for the particular crime, and this mental element may be either intention to do the immediate act or being about the consequence or (in some crimes) recklessness as to such act or consequence.
See Rex v. Crowe [1941] 4 DLR 82 (N.S.C.A.):
Perhaps it may be defined most precisely as the mental element necessary to convict for any crime. ...
The truth is that the maxim about 'mens rea' means no more than that the definition of all or nearly all crimes contains not only an outward and visible element, but a mental element, varying according to the different nature of different crimes ... Hence the only means of arriving at a full comprehension of the expression 'mens rea' is by a detailed examination of the definitions of particular crimes, and therefore the expression itself is unmeaning.
See also Black's Law Dictionary, 10th ed. (mens rea):
The state of mind that the prosecution, to secure a conviction, must prove that a defendant had when committing a crime
I wouldn't say the definition you quote from Oxford (I personally can't find that source) is wrong, but it is imprecise and clearly open to incorrect readings.
The mens rea often only requires that the person:
- knowingly did a thing, or
- did a thing with a particular intention, or
- did a thing recklessly (i.e. with awareness of the risk of a particular outcome, yet proceeding anyway)
For other offences, like dangerous driving, the fault element is negligence rather than a state of mind.
E.g. it is enough that a person know that they're taking a thing that isn't theirs. That makes out the offence of theft. They don't need to know that theft is an offence.
E.g. it is enough that a person intended to kill a person, and intentionally did the act that killed the person. That makes out the offence of murder. The accused does not need to know that murder is an offence.
E.g. it is enough to knowingly leave an ice fishing hole uncovered and unattended. That makes out an offence under s. 263(3) of the Criminal Code (if someone gets hurt). One does not need to know that this is an offence.
E.g. it is enough to knowingly harass a goose. That makes out an offence under the Migratory Birds Convention Act and its regulations. One does not need to know that this is an offence.
Only for some very specific fraud or obstruction-related offences does the mens rea require an intent to conceal a particular crime that you're aware of.