Does a bridge always necessitates creating an intermediate token?

When one creating a so-called bridge in order to exchange a coin on one blockchain to a coin on a different blockchain, how does it work? Does it always necessitate creating an intermediate token issued by a DEX?

A scheme is:

user A:
send KSM → receive INTermediate token, 1:1


user B:
send SOL → receive KSM


user A:
send INTermediate token → receive SOL

In all the steps the tokens that get sent and received, sent and received from and to a smart contract of a DEX. Right?

Does it work this way?

Cross-chain bridge mapping an asset from one chain to another. This is cross-chain communication. There are many protocol implementations.

After mapping, you can do something, such as swap.

User A mapping KSM to Solana chain by cross-chain protocol.
Swap to SOL with some defi contracts on Solana chain.
User A receive SOL on Solana chain or mapping SOL to kusama chain by cross-chain protocol.

The bridge is responsible for transmitting messages from one chain A to another B, and verifying the messages at B to ensure that the messages can be trusted. This can be implemented through the light client protocol.

The asset mapping and asset swap is application based on bridge.

Very unclear explanation.