If you view the same e-mail in webmail or in Thunderbird, does the same thing happen?

If not, then this is a matter of how your mobile is displaying e-mails, not how they are sent.

On computer it looks OK, on thunderbird and on gmail, at least. However in mobile (and just in smartphones because on tablet that's OK) it has this problem. I want that people open my e-mails on a smartphone, on gmail, for example, and view the signature correctly and not with images as attachement.

I think this is common in the mail clients on phones. I frequently find graphics are presented as if attached in my Android phones.

It doesn't happen if the signature is just an image. But as it is in html it happens. There's no way to have an html signature that looks fine on mobile?

I suggest experimenting with different HTML code in the signature. If you don't know HTML, then you can contact me directly by e-mail, sending me a text file with your signature, and I will look it over for any obvious problems.

Click on my name on the left to see my contact info.

I sent you a message with the code. Hope you can help me. Thank you!