Required conditions to be selected as intermediary node (Payment Service Provider) in Lightning Network? I bring up some important questions about terminology of the Lightning Network and how to select the intermediary node as the Payment Service Provider in a routing channel:

The following video (Link to video entitled: "How Bitcoin Lightning Channels Work") analogizes LN channels to abacus such that the channels are like rods including beads as bitcoin that slide from one end to the other, something similar to following figure (You can also find a similar article analogizing Lightning Network to abacus in this LINK):

 

Where, in this example, user A has 2 coins and user B has 1 coin.

This video also claims that:


  "There's no way to add more beads or take off a portion of the
  beads without taking the rod down all together."


And if this expression is correct, it means that the channel capacity is always equal to the channel balance, whereas, this answer (Link to answer) distinguishes between the channel capacity and the channel balance as follows:


  
  The capacity is the amount of Bitcoin provided to that channel.
  The balance encodes the amount of bitcoin which each channel partner owns in the channel. so the sum of the balances will always be the
  total capacity of the channel.*
  


Whereas, the video's description (i.e. "There's no way to add more beads or take off a portion of the beads without taking the rod down all together.) means that the capacity is always equal to the balance of the channel. 


  As first question: Which expression is correct?


Meanwhile, according to the video, the condition for selecting an intermediary as the Payment Service Provider is as follows:

If users A and B have a direct channel A-B, and users B and C have a direct channel B-C, such that user A has 3 coins in the channel A-B, and user B has 2 coins in the channel A-B along with 1 coin in the channel B-C. Then, assume user A wants to send 3 coins to user C. The situation is like the following figure:



We now want to investigate if user B can be an intermediary as Payment Service Provider, according to the example mentioned in the video:

Although, user B has a direct channel with user C and in general has 3 coins in both channels A-B and B-C ; however, user B can send only 1 coin to user C. Meanwhile, the capacity of channel B-C is 1 coin, whereas the transaction amount is 3 coins that is more than the capacity of the channel B-C. 

So, user B cannot be selected as Payment Service Provider (intermediary node) and users A and C have to establish a direct channel between themselves. 

If the above expression is correct, we therefore can define the conditions of an intermediary node as follows:

(1) The intermediary node has a direct channel with either destination or another node that has all conditions of an intermediary node.

(2) If the amount of transaction is n coins, the intermediary node has at least n coins payable towards/in the direction of destination.


  As second question: Are the above conditions for selecting the
  eligible intermediary node as Payment Service Provider true ?
Note that this video is made by shitcoin paddlers trying to spread FUD. If I remember correctly the further episodes are much much worse.


  "There's no way to add more beads or take off a portion of the beads
  without taking the rod down all together."


This is false. It is possible to both add and remove coins from a channel. It's called topping up or splicing. Having all your coins on one "abacus" is pointless. So who cares if you have 10 channels that you close one to pay something onchain? (Possibly using the same transactions to re-open the channel immediately.)


  this answer (Link to answer) distinguishes between the channel capacity and the channel balance as follows


I think you're reading that answer wrong. Of course capacity is different from balance as capacity is one number and balance is 2 numbers. Also note that "capacity" is a nice thing to talk about when talking about the total capacity of the whole Lightning network (as sum of all channel capacities) but the capacity of a single channel is kind of pointless. I'd argue it's actually more informative to define a channel as having two capacities: one in each directions. So your first drawing would have 2 + 1 as capacities. With this definition you can talk about the capacity of one channel in a certain direction as well as the capacity of a whole path from payer to payee over multiple channels (and a different capacity in the reverse direction). The total capacity of the whole network would still be the same.


  So, user B cannot be selected as Payment Service Provider (intermediary node) and users A and C have to establish a direct channel between themselves. 


False in two ways. 1) A can use this path for 1 BTC and send the rest over another path. 2) A and C don't have to connect to each other, they just need to have a few more channels. This is called Lightning Network after all.


  (1) The intermediary node has a direct channel with either destination or another node that has all conditions of an intermediary node.


Sounds correct.


  (2) If the amount of transaction is n coins, the intermediary node has at least n coins payable towards/in the direction of destination.


Yes. I'd even say "each intermediary node" (not "the") as according to (1) there can be more than one. But note that the payer always has the option of splitting "n" up into "n1 + n2", sending each over different paths.