You have saved your signature as an html file?

I'd use Thunderbird to create the signature by creating it as if it was an email message, but use File|Save As to save it as an html file, then set that file to be used as the signature in the account settings.

When inserting the image into the signature, visit its Properties and untick "Attach this image to the message". That should leave you with a bare URL.

You are sure your image is hosted somewhere that correspondents can download it from?

No, I copy-paste the code in the signature field.

I've set up a page on our website where every employee insert its personal info and gets an html code snippet to copy into thunderbird signature field.

I'm sure the html code is good and the image promptly available from the server.

I tried the same html signature with Apple Mail, the image is preserved as link and the message displays correctly (with the image) on almost every client (both mobile, web and desktop).

I'm testing the message on 20+ clients with litmus.com service

Try doing it my way; I think you'll see a tag modifier that supresses the inlining, but I don't have a reference to this right now. Then you can copy that modifier into your handcrafted signature code.

Awesome.

I tried what you suggested: composed a mail with a linked image and saved as html.

It turns out that if "Attach this image to the message" is unchecked then in the resulting html code a tag is added into the img element:

moz-do-not-send="true"

is inserted after the src attribute of the image.

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I then updated my signature html code adding the tag to the linked image and now the email is composed properly (without the image embedded) and displays fine on every device I tested.

The signature code can be copy-pasted into the preferences field or loaded from a file. What matters is that moz-do-not-send attribute is set true in linked images.

Thank you very much Zenos