The rule you're referring to is as follows:
'Due to' should always be used adjectivally, and never as a conjunction. For all other purposes, use 'owing to'.
Thus, you may say:
The cancellation of the service is owing to lack of interest.
The cancellation of the service is due to lack of interest.
Owing to lack of interest, the carol service has been cancelled.
But you may not say
Due to lack of interest, the carol service has been cancelled.
Kingsley Amis, in his book 'The King's English' had this to say
I have investigated the origins of this rule, and nothing substantial or satisfactory emerges. It seems to be just a rule.
Which does not mean that it can safely or praiseworthily be ignored, and I must confess that to my pair of ears or organs of grammatical fitness the exclusion of due to as an introductory phrase is justified.
According to this rule then, the sentence from your dictionary -
She has been absent from work due to illness.
is ungrammatical. The fact that it's printed in the dictionary is a reflection that, over the years, people have come to ignore the due to/owing to rule, and you now hear 'due to' used as a conjunction all the time (which can cause be annoying if you like the rule). Prescriptivism (language rules) is unfashionable in English these days, so it's not uncommon to find examples in dictionaries which break some 'rule'.