Japan had a "dominant party system" or did until 1993 or so. Since 1993, the LDP has lost power twice and only currently holds a supermajority by virtual of a coalition with another political party, since it can't manage that in its own right.
Sometimes a dominant party system is considered to have ended when lower political offices are lost by the dominant party, or free and fair elections begin to become very close even though it doesn't lose a full governing majority, but it ends no later when when it loses its parliamentary majority for the first time. But the mere fact that a party has a significant winning streak when there is some alternation of parties and there is some viable possibility that a party could lose an election, is not a dominant party system, it is just as multi-party or two party system with one party that is on average, stronger than the other. It is rare even in a two party or multiple party system that the parties are perfectly equally matched for any sustained period of time, even though parity is a natural "attractor" of the electoral system that tends to tug political parties in its direction.
Other countries have had such parties, which dominate politically even though other parties are not banned.
In Mexico, for two generations or so, it was the PRI (Institutional Revolution Party).
In the American South, during Reconstruction, this was the Republican Party, and from the end of Reconstruction until the Civil Rights movement, it was the Democratic Party. Arguably, the U.S. had a dominant party system during the Great Depression (during which the Democrats has immense supermajorities for many successive elections).
Most newly independent countries have a period of time when the party that secured independence, like the Congress Party in India, is a dominant party. Sometimes the dominant party becomes one leading party of many or undergoes a true schism into multiple parties, sometimes a dominant party system evolves into a true one party system.
Factions within a dominant party are really not closely analogous to different Western style political parties. Prior to 1993, it was in category 3 of the OP. Now, it may be in category 1 of the OP.
On the inside, the LDP doesn't look much different than a big tent western party like the U.K. Labour Party or the Democratic Party in the U.S. Japan has a narrower range of political opinion, but the internal process is actually less like an umbrella for multiple parties than the U.S. Democratic Party (since the U.S. party doesn't control membership or who it runs for office, and has no enforceable platform). The LDP isn't like an umbrella, even though, like all large organizations, it has internal factions.
Factions within a dominant party are more like factions within a one party state that has exceptionally liberal and meets high democratic standards relative to other one party states.
Dominant parties are particularly vulnerable in much the same way as one party systems, to corruption, because they face no short term electoral threats even if they behave badly (almost by definition).
In a multi-party system, different factions compete for support from members of the general public and debates over their disputes are public and there are few if any boundaries to the scope of acceptable political positions.
In a dominant or one party system, different factions compete for the support of the party leadership and party members (whose admission to the party is often regulated), key debates are often private and internal, and the party's organizational principles and history limit the scope of acceptable political positions.
Unlike a one party state, a dominant party usually eventually loses its dominant party status (as the LDP did not later than 1993 even though it eventually won re-election), although it often takes decades, and the mere possibility that this could happen creates some incentive for the dominant party to be responsive to the people.