I'm counting the same way. With stress in italics,

I am the an-i-mal but not the meat

sounds much better to me than

I'm the an-i-mal but not the meat

I should probably note that I read most of the lines with an approximately syllable-sized pause before 'but', and the animal line without that pause. The poem feels to me like it's written mainly in dactylls with some trochees and a final stressed syllable on each line.

Compare with

I am the play-ing, but not the pause

I'm the play-ing, but not the pause

I am the eff-ect, but not the cause

I'm the eff-ect, but not the cause

While I'm at this, how I read lines 9-11 as written

I am the waves, but not the sea

What-ev-er my sub-strate, my me is still me

I am the sparks in the dark that ex-ist as a dream

Which definitely break up the rhythm of the first half entirely, which is probably intentional, but particularly line 9 is awkward, which I didn't catch the first pass. If I was trying to keep that rhythm, I'd read it this way:

I'm the waves, but not the sea

What-ev-er my sub-strate, my me is still me

I'm the sparks in the dark that ex-ist as a dream

And be unhappy that "What'ver" is no longer reasonable English, even for poetry.