Note that if you operate under §135.267(b), that there isn't any specific duty time definition -- there are only regulations concerning the maximum flight time and minimum rest time in §135.267(d) (see this question for the definition of rest time). In a FAA Civil Penalty Order, it is noted that duty time is:
any time that is not a rest period.
If you're operating under §135.267(c), "a regularly assigned duty period" is also not defined, however, a legal interpretation in 2009 states that:
a pilot under a regularly assigned duty period "comes to work and ... goes home at the same time every day".
Now the question is, whether late passengers or being held up by ATC would be considered as affecting "the planned completion time of the assignment". The FAA has held in prior legal interpretations that:
delays caused by late passenger or cargo arrivals, maintenance
difficulties, and adverse weather are beyond the control of the
And in another legal interpretation:
delays due to air traffic control, mechanical problems, or adverse
weather could constitute circumstances that are beyond the control of
the certificate holder.
However, it is noted that these provisions only apply if the original schedule was realistic. For example, if it is typical that ATC will place aircraft into a hold for 1 hour and this is not taken into account, this could be considered an "unrealistic" schedule for the purposes of §135.267.
Note that if no initial "planned completion time of the assignment" was calculated prior to accepting an assignment though, then "being late" may not be a legal defence if taken to court. In 1995, a case involving Charter Airlines, Walker and Mort questioned whether it was a violation of §135.267(b) & §135.267(d) to accept a flight even when it was known that the cargo would be late to arrive.
Because it was found that there was no "planned completion time of the assignment", they were on duty when waiting for the cargo (and therefore not "free of all obligations") and they had accepted the assignment after it was aware that the incoming cargo would be late, the FAA issued civil penalties amounting to \$10000 for Charter Airlines, and \$2000 each for Walker and Mort.