Present Participle or Past Continuous?

What you have used is the imperfect tense, or as you call it, the past continuous.

It is formed from the simple past of the modal 'to be', and the present participle.

I was running, she was climbing, elephants were living, we were having dinner, you were going to college etc.

The imperfect implies continuity, unlike the past which refers to a snapshot in time.

Combinations of the auxiliary verb BE and the -ing form of a lexical verb are generally known as progressive or continuous forms. The tense of the auxiliary is the tense of the whole form. ... were living is therefore past progressive/continuous.

The progressive forms were once labelled as tenses, but most modern grammarians, for example Quirk et et al (A Comprehensive Grammar of the English Language, 1985.197) and Huddleston & Pullum( The Cambridge Grammar of the English Language, 2002.117) consider them to be aspects.