Why "I already said" is correct? I am confused with using past perfect and already

I have to improve my English, and I live in the USA.
I am confused with using past perfect and past simple after "already".
Is past simple correct after "already"?
I know, some people say it, but is this correct to say and write at work or university?

“Already” is usually used with the (1) Present Perfect and (2) Past Perfect. However, we can also use “already” with the (3) Simple Present, (4) Present Continuous and (5) Simple Past Tense.

Examples:

1. Have you already registered?
2. She had already read the news.
3. I already miss you.
4. We are already working on it.
5. They already left the city.

[EnglishStudyPage, adjusted.]

I'd add the caveats that

(a) using the simple present will involve a continuing state or iterative process, not a punctive event

• I already take vitamin supplements.
• *I already take the penalty.

(b) especially in the UK, the simple past is far less commonly used than the present perfect for a completed action / event

• They have already eaten / arrived / gone home.

• ?/??/*They already ate / arrived / went home.

• He'd only been gone two days, and I missed him already. [ongoing state]

• ?? I already missed the last post. [referencing a {notionally} fixed point in time]