Why do we need the MOESI/MESIF protocols?
MOESI allows more than one cache to have a copy of a block that is dirty (relative to main memory). MOESI provides all four states one would expect for the two binary traits present-in-only-one-cache/possibly-present-in-multiple-caches (M,E/O,S) and needs-to-be-written-to-memory-on-eviction/can-be-invalidated-without-writeback (M,O/E,S). (The invalid state excludes all four of those states.)

Under traditional use of MESI, if a cache miss hits a Modified block in another cache, the cache block would be written to memory, marked as Invalid in the providing cache, and marked as Exclusive in the requesting cache. Obviously, the M state could be transferred to the requesting cache (avoiding the writeback to memory), but the providing cache would still need to invalidate the cache block so multiple caches could not hold the block at the same time.

Avoiding writebacks can reduce the use of main memory bandwidth. (Alternatively, more flexible scheduling of writebacks which can distribute spurts of memory activity or exploit the lower cost of multiple accesses within the same DRAM row.)

The addition of the Forward state is intended to reduce the amount of coherence traffic by having at most one cache respond with data. (The Owned state provides a similar advantage but only if the "shared" block is dirty relative to memory.) On a cache miss, a request can be broadcast and any cache with the block in M, E, or F state can provide the data. With MESI, every cache with the block in Shared state would respond with the data.

(If the interconnect between caches was sufficiently loaded that there was buffering delay, this delay might be sufficient to check if the cache[s] at a network node had a block in a single-provider state [M, E, F], allowing a read request broadcast not to be sent further. Other, more practical optimizations are likely possible. "MESIF: A Two-Hop Cache Coherency Protocol for Point-to-Point Interconnects" might provide more information. [I have not read that paper, and my understanding of MESIF is even less than my understanding of MOESI.])