If at the end of the ride it would result in a new certificate or rating (ie. Private, Instrument, AMEL, etc.) it would be a checkride.
I think they really mean training failures as failures in a 121 or 135 environment where you're expected to pass each lesson. But you could probably include training "stage checks" like they do in 141 training.
The application last I looked at it wants an explanation of all failures anyways so be prepared to account for them.