For example, consider the sentence, “Ich springe hinein.” Is hinein considered to be the separable prefix of the verb hineinspringen or an adverb modifying the verb springen?
Yes, absolutely: hinein is an adverb, which may be used as a prefix in a compound. An adjective can also be used as a prefix, but nouns and verbs can't (at least with verbs).
The only distinction you (and me) have to learn is, that some verb-adverb pairs are so closely aligned, that both are written as one word for the infinitive form:
- schönschreiben, but schön schreiben
- wegfliegen, but weg reisen
Since the recent reforms, I always have to look that up in the Duden.