They both seem to be similar right?
They are not similar at all.
Partial Match = your keyword is part fo the phrase.
Alternatively, it can be just a part of the keyword in this case
dog or
food.
Example:
Keyword:
dog food
Partial match:
the best dog food store
dog treat
pet food
LSI = Latent Semantic Indexing
In common tongue it could be translated as "connected but not obviously". These words usually have completely different and meaning and even sound different than the "core" keyword but they are connected.
Example:
Keyword:
dog food
LSI:
carnivores
pointed teeth
canine
puppy growth
dog health
As you can see, your brain can easily tell that these keywords are all connected. Robots have troubles with this. That's why LSI was invented.
The best way to find LSI keywords imo is to identify the niche or sub-niche of the keyword. Then find related article on the Wikipedia. Wikipedia's content is filled with LSI keywords. Tha's how I fond the ones above (
https://en.wikipedia.org/wiki/Dog_food)
And we all know Wikipedia ranks for everything under the sun. Google absolutely loves it!
Wiki articles in general are dominating.
It's like a pyramid... keyword on top and then supporting words below it.
They are filled with useful information, yet they do not spam the keyword as is common practice among SEOs.