Does Coinbase tell me to send my private key from Electrum?

Tl;dr: What you have is

• Bitcoin wallet basics (where you manage your cash)
• The receiving address associated with your Coinbase account (if you allow Coinbase to take ownership of your cash in return for your IOUs)

You don’t have a private key (and you don’t need one, as long as you keep that seed phrase written down in a safe place and not on your computer).

Received 12 keywords

This is almost certainly a seed phrase, also known as a recovery phrase or backup phrase.

The master private key is generated from that seed phrase.

In a hierarchical deterministic (HD) wallet, that master private key is used to generate many private and public key pairs.

The most common type of Bitcoin transaction uses a public key to generate a Bitcoin address to which money can be sent.

And the hash of what I always thought was my Bitcoin key (starting with 17).

A string of approximately 32 or 33 letters and numbers starting with “1” is an old-school Bitcoin address that is still valid and widely used. This is a Bitcoin address, not a private key.

For some examples, take a look at some reference pages:

The same private key can be represented in different ways (encoding, display format).

The above should make you confident that you have a 12-word seed phrase for your wallet and an unrelated Bitcoin address for your Coinbase account.