I believe this is a configuration error in your php.ini. This is what is being generated in the ajax response, http://take.ms/nUAAT

And how to fix it? And can this be avoided on other sites?

You need to check this with your webhosting company as I am not too sure of the correct paths.

Good. Where is this site hosting is not very good. For work sites I use a different hosting.

Hm, you could try Cloud access or Inmotion hosting

Inmotion - really good hosting? Reviews on the Internet about it are not very good. And now it is very important for me to choose a good hosting

I would personally go with a hosting company that doesn't oversell or have much restrictions. Most hosting providers are very gracious for memory_limit and upload_max_filesize in the php.ini . Some hosting companies doesn't have any idea what these settings does and doesn't allow you to increase it (From what I have heard, ovh doesn't allow you to increase these settings)

I would personally go with either one of the following:

https://ccp.cloudaccess.net/aff.php?aff=4459
https://stackideas.com/inmotion-webhosting
https://stackideas.com/siteground-hosting

We do get an affiliate commission when you click on the links above but in all 3 of them above, I personally would go with Cloudaccess.

If you have spare budget or don't budget restrictions, consider getting a cloud vps instead. The only problems with cloud instances is that you have to manage them yourself. There are also several others,

https://linode.com
https://digitalocean.com
https://rackspace.com

Many thanks for the detailed answer. I will definitely use one of the hostings