I noticed something.. The website www.learnintouch.com has no API Key. I need to fix this first as it's most likely the cause of the issue.
So, now, after my fix on the code, the website www.learnintouch.com contains the API Key as in:
authorize: false // Do not auto log in the user but let him click on the "Login with LinkedIn" button
instead of no key at all.
And the website does not now display the LinkedIn button anymore.
I guess it is because LinkedIn is checking for the correct match between the domain name of the API Key to allow the displaying of the button, which is understandable.
What tricked me, was that LinkedIn, displayed the button, even without any API Key being used in the tag.
So I went to the page https://www.linkedin.com/secure/developer and changed the domain name related to the API Key, from www.thalasoft.com to www.learnintouch.com and sure enough www.thalasoft.com was not displaying the LinkedIn button anymore and www.learnintouch.com was. A consistent and sensible behavior. So, now I understand what was happening.
As a side note, I wonder if in the absence of API Key, LinkedIn should still offer the display of the button. But I'm no judge of that.
Thanks for your support !