The relevant Irish legislation defines those covered by the Visa Waiver Program as those with leave to enter the United Kingdom as a general visitor, child visitor, business visitor, sports visitor or academic visitor, within the meaning given to those terms in Part 2 of the United Kingdom Immigration Rules, that is granted by the competent authorities of the United Kingdom;
"General visas" cover short-term tourist and business visitors.
Other types of visa, for example Tiers 1-5, Indefinite Leave to Remain, Limited Leave to Remain, Post Study Work, transit visas, long-term student visas, join spouse or family reunification visas, are not encompassed.
Nationals of the countries listed, who are long-term legal residents in the UK, will still require an Irish visa but will have the visa fee waived as part of the programme.
The notes to carriers specifies:
It applies if you have a valid current UK visa in one of the categories covered, i.e. on the visa, the "Type of Visa" should be "C". Under "Remarks", one of the following should appear:
"C – visit"
"C – visit – general"
"C – visit – business"
"C – visit – academic"
"C – visit – sport"
If your UK visa category is listed above you should be covered under the visa waiver programme,
please ensure you follow the links above for full details.
If your UK visa category is listed as "C-visit family" then it is not eligible, and as you are required to make your visa application through your country of permanent residence.  lease see www.inis.gov.ie for details on how to make this application.
If under remarks on your visa it contains "C - Student - Visitor", or "C-Visit-Student", then it is not an eligible category under the visa waiver programme. However as an exceptional measure, should you wish to apply for a visa from the Visa Office, London, it will be processed free of charge.
Please see our website for how to apply, processing times, etc: https://www.dfa.ie/irish-embassy/great-britain/our-services/visas/visas-for-ireland/#
A list of frequently asked questions can be found at https://www.dfa.ie/irish-embassy/great-britain/our-services/visas/top-visa-questions/
this list may address some of your queries.
Embassy of Ireland,
114a Cromwell Rd
London SW7 4ES.