Oftentimes I see phrases such as this in songs and such:

僕は迷い込んだ夢の中

I always interpreted these as the first part all modifying the noun, i.e.
"Inside the dream I was lost in," here. However, when I asked a question on
this particular song, a user translated this with the subject as separate from
the modifier, i.e."I'm within an astray dream." This came as a shock to be,
since I've always thought of lines like this the first way.

Are both interpretations correct, or is one of us wrong?

Thank you!
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question
particle-は nouns は-and-が modification
For phrases modifying nouns, is the subject considered part of the modifier or separate?
It's unnatural to put the 僕は into the could-be modifying clause, i.e "Inside
the dream I was lost in" (In that case, it's usually 僕が) unless the dream
world is shared with several people and you were lost there unlike others.

So, it's more natural to interpret it as 僕は 夢の中 迷い込んだ ("I wandered into a
dream" or "I'm lost (in a maze) in a dream") with the word order inverted or,
as someone said, 僕は（自分が）迷い込んだ夢 の中, though this is an unnaturally recursive
expression unless there's a particular reason to use.

That said, only the author can know how to interpret the lyrics.
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Your parsing is not quite right. Literally, 僕 **は** 迷い込んだ夢の中 means "I am in a
dream I strayed into." Note that it's basically a full, regular sentence
except that the last verb だ is omitted. I can say this because, as a general
rule, は is a "topic" marker and thus rarely appears in a relative clause. This
is a [basic rule of relative
clauses](https://japanese.stackexchange.com/a/14550/5010):

> But before we can relativize them, let's replace the topic particle は with
> the case particle が. Why? Because relative clauses in Japanese don't contain
> topics, so you need to use the version with が to turn them into relative
> clauses.

In other words, this sentence is 僕は夢の中だ ("I am in a dream") with a short
relative clause 迷い込んだ.

The subject of 迷い込んだ is 僕, which is the same as the main verb, so it's
naturally omitted in this case. You can explicitly specify the subject of the
relative clause with **が** like: 僕は僕が迷い込んだ夢の中だ or 僕は彼女が迷い込んだ夢の中だ ("I am in a
dream she strayed into.")

By contrast, 僕 **が** 迷い込んだ夢の中 means "middle of the dream I strayed into" (a
noun phrase without a main predicate; sentence fragment).

Related:

  * [How can we determine the subject of 「彼{かれ}が来{こ}ないことを知{し}りませんでした。」?](https://japanese.stackexchange.com/q/30698/5010)
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