Is something "interpreter" or not depends on context of discussion.
From purely abstract view interpreter can be defined as any intermediate program present in runtime which dynamically translates program code written in one language to a target code of hardware/software of other language. Think about runing java bytecode on x86 hardware, or running Python on CLR VM what exactly IronPython is. In this view every virtual machine is an interpreter of some kind. As it is program present in runtime it clearly differs from static compilers or hardware implemented VM-s.
Now there are many different ways to achieve this functionality where accent is on "dynamically" and "present in runtime".
In discussions where implementation of VM matters, people make clear distinction between "classical" interpreter and JIT-ed one. Classical interpreter is something which for every instruction of hosted program emits routine of target code. This design is simple to build, but hard to optimize. JIT-ed design reads bunch of instruction of original code, and then translates all those instructions to a one native compiled routine. So it "interprets" faster. It is like micro static compiler within VM. There are many different ways to accomplish behavior labeled as JIT, and then there are other approaches like tracing compilers.
Modern VM's like CLR, HotSpot and J9 JVM's are even more complex than to be tagged with simple labels as JIT or Interpreter. They can be at a same time static compilers (AOT execution), classical interpreters and JIT-ed VMs.
For example CLR can compile code Ahead-Of-Time (static compiler), and store native code as bunch of more or less excutable files on disk to be used for faster future startups of hosted program. I believe "ngen" is AOT process used in windows for this functionality. If AOT is not used CLR behaves as JIT VM.
J9 and HotSpot are able to switch in runtime between purely interpreted execution or JIT-ed on depending of code analysis and current load. So it's is quite gray area. J9 even has AOT functionality similar to CLR.
Some other VMs like Maxine JVM or PyPy are socalled "metacircular" VM. This means they are (mostly) implemented in a same language they host (Maxine is JVM written in Java). In order to provided good code they usually have some JIT like behavior implemented in host language which is than bootstrapped and optimized by a very low, close to machine, interpreter.
So actual definition of interpreter varies on context of discussion. When labels like JIT are used then there is clear accent of discussion to an implementation details of VM being discussed.