Explanation of "Timestamp Server" section in original bitcoin paper
Is it saying ...

I think you have it right except that for 2.2. the whole new block is broadcast not just the hash.
Actually &quot;broadcast&quot; isn't exactly the right term really. I believe the node transmits the new block to a small number of peers (e.g. ten or so other nodes). These other nodes then pass on the new block if it looks OK to them. Eventually the new block reaches all nodes in the network.
In a comment below, Pieter Wuille has clarified some of the details, &quot;Blocks are announced to all peers. Those peers then fetch the actual block from (typically) the first peer that announces from them.&quot;
I haven't found an explicit detailed description of the exact sequence of network actions, but if you read The Bitcoin Wiki's Protocol documentation you'll see that there are many message types and one of them is inv  (inventory)

Allows a node to advertise its knowledge of one or more objects. It can be received unsolicited, or in reply to getblocks.

The inv payload includes an array inv_vec[] which contains a code for a type of data available (e.g. block) and a hash of that data.
So the recipient of that message could presumably see that the sender has one or more blocks that the recipient doesn't and could presumably then issue a getdata message to obtain that new data. As I said, I haven't yet found a detailed explanation of the exact sequence of operations but I imagine the reality is along these lines. So my description of this as broadcasting new blocks is a simplification.


There is little information online trying to research the concept of &quot;timestamp server&quot;

As you evidently know, but others might not, what Nakamoto was describing, under section 3. Timestamp Server, is an activity that we would now refer to as mining.