[SqueakyRat]

I see.
I think I should have explained some Japanese customs before I wrote the sentences.
Most Japanese names have meanings. Have you heard Japanese has letters called kanji? Each kanji has its meaning. When we write our names, we use kanji, so usually our names have its meaning.
For example, Ichiro.
This name means [the first son of the parents].
The first son of the parents is [chohnan] in Japanese, but this can’t be a name, but a common noun. (Ichiro can’t be a common noun.)
Taro has the same meaning.
Jiro means the second son of the parents.
My family name is Tokoro, and it means place.
Yuri is lily.
Do you see why names can have meanings,now?

Oh! I get it now!
Thank you, while I haven't really started learning Japanese yet, I vaguely knew about this, I just couldn't draw the connection. This helps a lot to understand your sentences.

OK.
Then, how about this?:
The name means a mind. = Der Name bedeutet eine Verstand.

Just drop [eine] and you're good to go. (It's also normally [ein], when used together with Verstand.)

I have written the word spirit meaning like “spirit (of collaboration)”, “spirit (of fair play)” or “spirit (of friendship)”.
I think I shouldn’t have written the “a”.
I’m sorry. My English is poor.

It wasn't your English, the first thing I think of when I hear [Geist] in German is a ghost. Although there are other uses, in some cases it can also be used instead of [Verstand] and yes, now that I think about it, we use the word [Teamgeist] which literally translates as team spirit.

In that case your initial sentence was almost correct actually, just make it [Teamgeist], instead of just [Geist], and again drop [ein].

Hmmmm…..
Does your sentence mean “He enters somewhere under the name of Offenbachs”?

I thought “enter up” means “fill in” , “put down one’s name down”,
“make an entry” or “sign”.
I imagined someone check-in a hotel or something and write the name and some address at the reception, when I wrote the sentence.
This online dictionary says eintragen means to sign.
Translation eintragen - German-English Collins dictionary - Reverso
Can’t I use this word in the sentence?

I see. I thought you meant 'He enters on behalf of Offenbach'. And that's what the German sentence I gave you in my previous comment meant.

Since Offenbach is also the name of a German City and the first thing that came to my mind it kinda made more sense to me.

In your case the correct sentence is [Er trägt sich unter dem Namen Offenbach ein], using [eintragen] is correct as well, I wasn't thinking about a hotel but a team or a club of sorts, in that case [eintreten] would be the verb you should use, as I did in my previous sentence. (tritt...ein.)

[Er ist unter dem Namen Ballack bekannt.]
This sentence doesn’t have “den Namen”, but has “dem Namen”.
So this seems to be good for an example with “dem Namen”!!
Now, I need an example with “den Namen”.
How about this?;
I remember the name. = Ich merken den Namen.

merken = memorize

erinnern = remember

[Ich erinnere mich an den Namen.]

I see…
I need an example with “die Namen”.
If you have any, could you tell me please!

No problem. Instead of [Diese Namen wecken Erinnerungen] just say [Die Namen wecken Erinnerungen]. They're both correct.

I need a sentence including “die Namen” here.
Then, how about this?
He gives me the name. = Er gibt mir den Namen.
(I haven’t studied the past tence,yet.)

Huh? Your German sentence is correct, but I though you wanted a sentence with [die Namen].

[Er gibt mir die Namen] -> [He gives me the names]

Please!! Give me an example sentence including “den Namen”.

Here are a few I just thought of.

Den Namen habe ich noch nie gehört. (I've never heard that name.)

Wie findest du den Namen? (How do you like this name?)

Er verrät der Polizei den Namen des Täters. (He gives the culprit's name to the police.)

Er schreibt die Namen auf das Papier.

Correct.

I see.
German seems to be more difficult than I expected.

Well, it's certainly not easy, but I do think my own confusion from my last post caused you more confusion than it should have. I hope things are a little bit clearer now.