Does an overhaul reset the 'life limit' of a turbine engine?
All nations operate with very similar rules. For the US, the short answer is nearly all turbine powered aircraft operate under part 119/121 (air carriers) rules. Each company has maintenance and operations specifically tailored and tied to their operating certificate and so it depends on the approved procedures of that companies operating certificate.

In your example, no mention is made of the part being "rebuilt", therefor the use of the part will continue where the previous use left off. The card says it is a "life limited" part so it's use will run out according to operating hours, calendar time from first installation, or number of operating cycles, etc. The parts life span is set by the mfg or operating certificate rules.

For example if a part  has a 7yr calendar "life limit" (like some seat belts) and the part was installed January 1, 2015 then removed 10 days later, the part would become worthless Jan 1, 2022 even though it was used less than 1 month - the clock starts ticking from it's first use.

This card says the part has a 15yr calendar limit which will usually be in addition to other limits. It will either be limited from it's first date of use, or it's date of manufacture (whether the part is used or not). Once again, exactly how it should be interpreted will be based on the mfg or operating certificate rules. 

Most life limited part life spans are TIS (time in service) and therefore expire after a certain number of hours of operation. If TIS for the turbine blade is 5000hrs and airplane "A" used it for 4000hrs, it could remain on the shelf indefinitely - be re-installed on airplane "B" and used for an additional 1000hrs. The card is designed to keep track of the dates and how long the part has been used - TIS, calendar, cycles, or whatever.

[I'm guessing for this particular part, but for many critical parts on helicopters and turbines a combination of TIS and calendar are used. So it would be something like 5000hrs TIS or 15yrs calendar from date of mfg - whichever comes first. Even if it sat on a shelf and was never used, it would have to be destroyed when the 15yr limit was up.]

NOTE: Due to a problem of expensive time limited parts being counterfeited, re-sold, or re-used, the FAA requires companies using life limited parts (once the service life is up) to make the part unusable by returning to a mfg, defacing, or destroying it.

Parts

On any US registered aircraft, "overhaul" and "rebuild" have very specific legal and maintenance meaning. "overhaul" per FAR 43.2(a) means "Disassemble, cleaned, inspected, repaired, reassembled, and tested to service limits", whereas Rebuild FAR 43(2)(b), 43(3)(j) means "disassemble, clean, inspect, repair, reassemble, and test to new part tolerances".

"Overhaul" means repair or fix the old part and does not change the wear or life of a part. 

"Rebuild" means to make a "new" part out of the old one and the part's life is started over - "zero time". The part is treated as if it is a brand new part and many mfg even stamp a new serial number on the part. 

Entire Engines

The above FAR's relates to "parts". For an entire engine, the FAA's AC-43-11 (Reciprocating Engine Overhaul Terminology and Standards) goes into great detail on what the terms mean and who may perform the work.

Why would AC-43-11 (entire engine) be limited to reciprocating engines? Because nearly all turbine engines are operated under Part 119/121 (not 91 or 135) operators. Part 119/121 receive FAA approval on a company by company basis and each company has maintenance and operations specifically tailored and tied to their operating certificate. 

When a part 119/121 company has their operations approved, the FAA's standard practice is to hold them to all existing FAR's unless they show a "as good" or "better" way of maintaining the same level of safety. So, the spirit of AC-43-11 would be applicable unless some exception was given.