Wouldn't it be more correct to say just "store of value" (short-and-long-term)?
A store of value relates to price stability. Something that stores value is something that is worth approximately the same tomorrow (or in a week, a month, a year or in 3 years) that it is today.
A medium of exchange is something that is accepted by the public for goods and services.
One example of something being a store of value, but not a medium of exchange is a refrigerator. In general a refrigerator will not gain or lose any substantial value over a few months. However it would not be realistic to offer someone fourteen refrigerators to buy a car as the other person would have little use for that many refrigerators even though they hold their value.
You are correct that every medium of exchange is a short term store of value, but this does not do any good for something to be considered a currency. If something is a bad long term store of value (but good short term store of value) then people would not wish to invest in that currency as it could be worth a lot less when it needs to be spent. The same is true for savings and being paid a certain way.
Generally for something to be considered a store of value it needs to be a store of value over a longer period of time.