If we follow the pattern of 'cat lover', is it correct to say 'run lover'?
If I use Google translator to Spanish (my mother tongue), 'run lover' sounds more like a shout you'd say to your lover to start to run.
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I think Mike C. has a great answer, but if you'd prefer a noun version over an adjective, I would call the person a running enthusiast
A person filled with or guided by enthusiasm.
Enthusiasm, of course, being:
Intensity of feeling; excited interest or eagerness.
So a running enthusiast is one who is excited about running or eager to run.
In Spanish, the same form is used for both the infinitive and the gerund. So when translating to English, you need to be able to distinguish between the two. Here, "correr" is a gerund, not the infinitive, so it should be translated as "running". However, in English the same form is used for both the gerund and the present progressive, so "running lover" can be ambiguous as to whether "running" is a gerund ("amante de correr") or present progressive ("amante corriendo"). Using the term "enthusiast" makes the gerund interpretation more prominent. The term "aficionado" can also be used in English: "running aficionado", although that can be ambiguous as to whether it's someone who likes to run, or someone who likes to watch other people run. Using the noun form "runner" also avoids the ambiguity between gerund and present progressive, and then an adjective such as "avid" can be added to describe what sort of runner they are.
One who is addicted to running.
Example from I'll Meet You at the Finish!, written by Chris Pepper Shipman and published by Life Enhancement Publications (1987), page 65:
One such woman complained, "My husband has become a 'runaholic'. He constantly talks about his running time, his equipment, people he has seen doing unusual things while he was running and on and on and on. […]
You could try "amateur" as an adjective, e.g. "she's an amateur runner".
Noawadays that mostly means that, "she's not a professional runner", but it originally meant that, "she runs for the love of it, not for money".
An amateur, from French amateur "lover of", is generally considered a person who pursues a particular activity or field of study independently from their source of income.
I mention that because it includes some of the "love" that you were asking for (if you accept a word's etymology as having any meaning, which maybe some people don't).
Also, "amateur" is often (idiomatically) used of sports.