Required conditions to be selected as intermediary node (Payment Service Provider) in Lightning Network?

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:

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 ?

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