Why does Bitcoin use both a transaction ID and a witness transaction ID?

The useful part

While learning Bitcoin transactions and SegWit, I came across the distinction between the traditional transaction ID (txid) and the witness transaction ID (wtxid). I understand that the txid is the double-SHA256 hash of the serialized transaction excluding the witness data. Bitcoin needs both identifiers instead of simply replacing txid with a hash that always commits to the witness data.

How it works

• For example, suppose a SegWit transaction has the same inputs and outputs but different witness data.
• Is the separation primarily a backwards-compatibility decision, a consequence of how SegWit fixes malleability, or are there deeper protocol reasons?
• andd wtxid commits to the transaction including the witness data.
• How does this distinction affect the mempool and transaction relay?
• Why do transaction inputs continue to refer to previous outputs using txid:vout rather than wtxid:vout?

What to take from it

Why does the witness merkle tree in a block use wtxids while the normal merkle tree uses txids? I am particularly interested in the historical/design reasoning rather than just the definitions of txid and wtxid. References I've been reading so far include BIP141 and the Bitcoin transaction/SegWit sections of Mastering Bitcoin.

Details worth keeping

SegWit introdduced wtxid partly to solve transaction malleability issues. The txid remains unchanged while the wtxid changes. I would like to understand the design consequences of this distinction...