I would call it an adjective in both.
I'm wondering if "bored" in either of the following examples is a past participle used as an adjective ( SUBJECT+BE+ADJECTIVE) or a part of a passive voice sentence ( SUBJECT + BE + PAST PARTICIPLE)?
1. I'm bored.
2. I'm bored by this lesson.
MY OWN GUESS: Can we say in the first example it's just a past particle used as an adjective, but in the second it's a past participle used in a passive sentence, just because in the second example it's followed by a "by phrase"? in other words, if we consider "bored" simply as a participial adjective, can't it be followed by a "by phrase"?
Your help is greatly appreciated.
It's an adjective in the first. I'd say that the second sentence is a typical passive construction with by = agent. This lesson bores you - you are bored by this lesson.
It doesn't really matter, does it?