The lag in bounded staleness refers to rate in which data is replicated to secondary regions in a distributed database. Unlike other relaxed consistency models, bounded staleness enforces the amount of time or number of updates in which data between the write region and secondary replicas are not consistent. When the data approaches the staleness window, bounded staleness will throttle the number of writes in order to allow replication to catch up.
The consistency guarantees are written this way because the behavior appears different depending on where the readers are and where the writers are.
For #1, this applies to scenarios where both the writer and readers are in the primary write region. (The way this is worded is a bit confusing. Will get that updated). Another thing to understand about this scenario. Reads for bounded staleness are a minority quorum (2 replicas). This is done to ensure the most recent data is read in the write region. When data is written it is a majority quorum (3 replicas). A replica set is 4 replicas in a region. When data is read from the two replicas, the LSN for each replica is compared. If they match the data is guaranteed to be the most up to date and is returned. If they do not match, the replica with the higher LSN is returned because it is the most up to date. This is great for ensuring consistent data, but the cost of this is 2x that of Session or weaker consistency because you are reading from two replicas. (PS, this is the way Strong consistency works as well for reads. The difference for Strong consistency is that data is not committed until it is written to every regional replica. This is why write latency is so slow when using strong consistency).
For #2, that is correct. When the writer is in one region and the reader in another, the reader will see consistent prefix (reads are in the order in which they were written).
For #3 and #4, in a multi-region write scenario, you can have more than one client instance reading and writing in a region. When in the same region they will see consistent prefix, when in different regions they will get eventual consistency.
There are some videos which may help here. Key to understanding consistency models is understanding CAP and PACLC theorem. The segment in this presentation from BUILD in 2019 does a good job explaining how distributed databases handle all this. https://youtu.be/fOQoQnQqwwU?t=1133
And this is a short video explaining each of the consistency guarantees. https://www.youtube.com/watch?v=t1--kZjrG-o