OP's examples use even in the sense as per the full Oxford English Dictionary definition #8...
Used to convey that what is being referred to is an extreme case in comparison with a weaker or more general one which is stated or implied in the adjacent context.
Prefixed to the particular word, phrase, or clause in which the extremeness of the case is expressed.
Now the prevailing use of the word in English.
1: Alice's cooking is so bad that even her dog doesn't want to eat it
...we're being presented with her dog as the most extreme example of a "reluctant diner" (nobody wants to eat her food, not even her dog).
But with the (extremely unlikely) version...
2: Alice's cooking is so bad that her dog doesn't even want to eat it
...the implication would have to be that her dog's attitude to the food is "extreme". Stereotypically, dogs are extremely enthusiastic about any food on offer, so for her dog to not actually want the food at all is an extreme case.
That second version would be an "unusual" thing to say, but it might make sense if we suppose the context is that the dog does in fact eat her cooking - but reluctantly and without enthusiasm (because it's told to eat, force of habit, or just to avoid starving to death - it doesn't actually want to eat it).