What is the exact definition of a "crew duty day"?
How do you define a crew duty day? When does it start and when does it end?


In the most simplistic terms, a crew's duty day can be defined in two parts: the "flight duty period" (FDP) and the "flight time". 

The FDP begins when the crew "clocks in" for their shift. There might be administrative duties to accomplish prior to flying, so "flight time" has not yet started. The FDP ends when the last flight that they operate blocks in. The Flight Time begins and ends with each Block Out to Block In, and there might be several individual flights within a single FDP. Once the FDP is over, they might still be required to attend to admin duties, but their FDP "clock" has stopped ticking. Their "rest" clock has not yet started, just because their FDP has ended. (Some of these terms and definitions will be expanded upon later.)


  Also what are the legal limits (in other words how long can a crew duty day last in terms of hours)?


This is an extremely complicated question, and the rest of this answer will be devoted to this, in conjunction with your other sub-questions.


  [Are there] standardized set of rules?


Not necessarily rules, per se... but the ICAO does have a lot of documentation about duty/rest, fatigue management, and the effects on safety. Although the ICAO cannot publish laws, most of it's member nations write their aviation laws in close conjunction with ICAO's recommendations. These nations have ratified treaties stating that they will "abide by" most of the ICAO's recommendations.




  does each country define its own rules?


Yes, each regulatory body has its own laws for determining duty/rest requirements. Most are very similar, and based on the ICAO's body of documentation and research.

Narrowing this down even further, each carrier develops it's own policies and procedures that are based on (and approved by) it's parent nation's regulatory body.




  What happens when you fly say from the US to Australia? Which set of
  rules prevail?


You will follow the policies of your air carrier, which are enforced by your carrier's country of registration.




  Let's start with the US.


Ok, within the US, most of the rules regarding duty/rest for Part 121 Air Carriers are enumerated in Part 117: Flight and Duty Time limitations and Rest Requirements: Flightcrew Members. 

There are some "All Cargo" duty/rest requirements under Part 121 Subpart Q (Domestic), Subpart R (Flag), and Subpart S (Supplemental); however, carriers can chose to use these 121 Subparts OR Part 117, but not both. Once they make that choice (listed in their GOM/OpSpecs and approved by the FAA), they are stuck with it.




  What is the exact definition of a “crew duty day”?


As noted in the comments of the question, this is not a legal term within the US FARs. Under Part 117, we find the definition for these related terms:


Calendar day means a 24-hour period from 0000 through 2359 using
Coordinated Universal Time or local time.
Duty means any task that a flightcrew member performs as required by the certificate holder, including but not limited to flight duty period, flight duty, pre- and post-flight duties, administrative work, training, deadhead transportation, aircraft positioning on the ground, aircraft loading, and aircraft servicing.
Flight duty period (FDP) means a period that begins when a flightcrew member is required to report for duty with the intention of conducting a flight, a series of flights, or positioning or ferrying flights, and ends when the aircraft is parked after the last flight and there is no intention for further aircraft movement by the same flightcrew member. A flight duty period includes the duties performed by the flightcrew member on behalf of the certificate holder that occur before a flight segment or between flight segments without a required intervening rest period. Examples of tasks that are part of the flight duty period include deadhead transportation, training conducted in an aircraft or flight simulator, and airport/standby reserve, if the above tasks occur before a flight segment or between flight segments without an intervening required rest period.
Rest period means a continuous period determined prospectively during which the flightcrew member is free from all restraint by the certificate holder, including freedom from present responsibility for work should the occasion arise.




From there, it can get extremely complicated and an airline's crew scheduler probably could not do their job without the aid of a computer. Flight crewmembers are restricted to a certain number of hours in a duty period, and there are limits to cumulative hours as well:


total Flight Time cannot exceed:
-- 100 hours in any 672 consecutive hours or
-- 1,000 hours in any 365 consecutive calendar day period.


And then...


total Flight Duty Period cannot exceed:
60 flight duty period hours in any 168 consecutive hours or
190 flight duty period hours in any 672 consecutive hours.


They are required to have a certain number of hours of "rest" in between these duty periods, and those rest hours are required to be in a certain type of facility (if they are not at home). 

The amount of time in the cockpit for a flight is also determined by what time of day you start, whether or not there is an on-board relief crew, whether or not you have crossed more than 60 degrees of longitude (and if you HAVE, how long have you been within the new "theater"), and several other factors.



So, finally, back to your question:

What is a crew's "duty day", and what are the legal limits? The answer is so complicated, that the Air Line Pilot's Association has published a 60-page document (updated several times) with 128 scenario-based Q&amp;As just to help "clear it up". 

ALPA's FTDT Limitations and Requirements (May 2015)