I noticed some people using the terms as if they were synonoms.
For instance, in the same scenario, I heard "add this function to the lisp image evaluating it" and "eval this function into the Lisp interpreter to use it later".
However, I am not sure the use is technically precise. Thus, the question.
These are two orthogonal concepts. Let’s start from the usually comprehensive Common Lisp Glossary:
Lisp image n. a running instantiation of a Common Lisp implementation. A Lisp image is characterized by a single address space in which any object can directly refer to any another in conformance with this specification, and by a single, common, global environment.
So the key idea is that an image is a set of mutually referring Lisp objects (functions and data) that can be “called” or “accessed” during the execution of a program.
The way in which a Common Lisp program is executed depends instead from the way in which a system is implemented. It could be executed by compilation in machine language, for instance, or through some form of interpretation (or even a mix of the two). So a Lisp interpreter is just a particular way in which an implementation is done (and in the current Common Lisp systems there are many different ways to implement the language).