That's right, but I don't think it has a lot to do with livingness, or am I wrong?
Because a fork doesn't live either and it's still "gaffeln" and not "gaffelt" or "gafflet" when definite.
Same goes for "child" which is "barnet" and not "barnen" when definite. And a child does live.
See my answer to EllieLemke.
Sometimes studying languages that have strong similarities can be more difficult this way: a difference in meaning that is neutralised when translating into a language that is very different can be kept when translating into a language that is very close.
kött uses the SHORT ö vowel, so rather than shiiiiirt (dragging out the main vowel) it's more like shirtt (clipping the vowel a litle short). Still, to most English speakers the vowels sound similar but with slightly different mouth shapes. The English [ɜː] in "shirt" is produced with little effort, but in svenska it seems to me [œ] for short ö and [øː] for long ö are pronounced with more intense, forward lips, a bit more work for the mouth