I integrated my JIRA project with TestLink using http://blogs.quovantis.com/step-by-step-integration-of-jira-with-testlink/.
However, there is no way on JIRA to understand that TestLink is integrated successfully and vice-versa.
As with TestRail, when integrated with JIRA, I could see the "TestRail" tab on JIRA dashboard. Also, in TestRail while creating Test cases I could see separate field "References" to add JIRA ID reference. Upon adding reference, test case ID gets reflected in same JIRA ID as well.
Check the image for reference.
How to check whether TestLink is integrated with JIRA?
From what I see on the page you linked there is no interaction with JIRA in regards to connecting the two platforms but there is a clue to how you can tell:
You can directly refer to JIRA tickets by providing the ticket number. Otherwise click on the link ‘Access To Bug Tracking System’ to access JIRA. For first time access of JIRA via Testlink, you will be redirected to you JIRA Dashboard, prompting a login in order to create a new ticket.
Also, during setup of the link there should be a "Test Link" option that should let you know if the link is successful or not per the article you referenced. Since TestRails is an add-on that is why you're seeing it in JIRA as a menu option.
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