My service provider sent the information to me. Do I have to get rid of the X stuff?

You did not answer my question.

If X is off, isn't that enough?

Where can I set the email to be sent by smtp instead of the script?

Ok,
and did you show this to your provider and ask him how to get rid of this 'X-....' stuff in the meantime?

mailtester:

Okay,
my thinking may be a little different, but before ADDING something new (in this case SMTP) to prevent something else,
I'd first try to find out if the EXISTING configuration can't be changed some way ... ;)
So if we have a look into the details, we see

To: [email protected]
Subject: =?UTF-8?Q?Izgalmas_k=C3=A9r=C3=BCnk_igazold_vissza_az_e-mail?=  =?UTF-8?Q?_c=C3=ADmedet_a_Elittag_-_k=C3=B6z=C3=B6ss=C3=A9g!?=
X-PHP-Script: elittag.hu/system/handlers/actions.php for 84.1.50.62
X-PHP-Originating-Script: 1032:class.phpmailer.php

and lot of other stuff starting with 'X-.....'

That's reason the mail-tester is giving that bad result.
So when do we get this additional X- ... stuff?
Normally, if mail.add_x_header is set to ON

So, the next thing I would do is installing the System Info component or running the standalone ossninfo.php?view=phpinfo
to verify the current setting. On my site it looks like

in YOUR case it's most likely set to On.
And if so I'd ask my provider which way to turn it off and see if that makes a difference

I wanted to add my few experience here. In last few days I had similar situation where one of my domain sending messages into spam not mailbox. My mail-tester score was 10/10 but still goes to spam, my IP was brand new never been in blocklist.

The problem is many domains and new servers/ips have not enough reputation and email providers sending mails to spam. There is nothing you can do to fix it.

The only luck is to change to a dedicated IP or find a SMTP that provides a good IP as Balamurali and it worked for me as well , I will say same thing mentioned by Z-men that its not a OSSN issue.

Use a good SMTP provider and the emails show in inbox.

https://www.mail-tester.com/test-0eaihwccf
I registered on the elittag.hu page to the given email address, then I checked the points on the tester page.

Please tell us more details about which way you did that test
and the exact response from mail-tester.com