if the device is listed with a 5 year lifetime or warranty, and we continue to repair and refurbish devices well past this date, does this change the lifetime of the device?
what would the most appropriate definition for "lifetime of the medical device" be?
what is the best way of determining the lifetime of the device?

One example definition is the one from the new IEC 60601-1 standard:

EXPECTED SERVICE LIFE
maximum period of useful life as defined by the MANUFACTURER

Keep in mind that, if the manufacturer specifies a 5 year lifecycle (it´s the common word used, not lifetime), and you act on the device past this date, for some regulations you´ll be the new manufacturer of the device and will have to comply to regulations (and also define a new expected service life)

The best way of determine the service life of the devices is through reliability analysys and tests.

The lifecycle of the device is everything from the design and development of the device to the decommisioning or proper disposal of the device.

The lifetime of the device could be how long you expect the device to be functional (expected lifetime) A device may last longer than its lifetime but your organization may continue to service it (usually for a fee).

From ISO 14969 section 7.1.3, the rational for the Lifetime could be as follows:

a. Shelf life of the medical device
b. Expiry date of the medical device
c. Number of cycles or periods of use of the medical device
d. anticipated material degradation
e. stability of packaging material
f. for implantable devices, the residual risk that results from the entire period of residence of the device in the body
g. for sterile devices, the ability to maintain sterility
h. organization's ability to support service
i. spare parts cost and availability
j. legal considerations including liability