Question: As a Buddhist, when I attended a funeral service of another faith, I noticed there was a lot of praise and thanks to the deity believed in.
Answer: This is to remind followers present of their faith, while impressing upon others of it. However, according to other teachings, deities with the attributes believed in might not exist: https://thedailyenlightenment.com/2018/08/can-faith-in-any-deity-help-the-dying. It is puzzling why there should be grieving at such wakes if there is belief that the deceased is now with the deity happily. (There is also the question of why the deity would ’cause’ deaths that grieve survivors.)
Q: I wanted to do silent Nianfo (practice of Amitabha Buddha’s name – ‘Amituofo’) but the deceased was very devoted to his faith. So, I was concerned that should his consciousness, if still around, might be confused. Though we are of different faiths, I hope he will have a good rebirth.
A: Yes, to be respectful and not disruptive at funerals of different faiths, following silent words of guidance in your mind thought ‘aloud’ sincerely, silent Nianfo should be done. You should mentally go through the sequence of practice list at https://purelanders.com/2016/07/19/what-should-be-chanted-during-after-a-funeral, adapting where practical due to constraints.
The truth is, without doing the proposed above, the deceased, if still around, is likely to be more confused – because if the deity believed in does not exist, or is not as powerful as believed, there will be no one coming to guide him to a better realm, while he will be at a loss on what to do. This means the chances of having a good rebirth are lessened. If he does have a good rebirth, it will be based only on his good karma, which might not be good enough.
Q: Not doing Nianfo, I listened to what was said during the service. I found myself going through an emotional ride as tears started to flow as people started to relate good deeds of the deceased. I tried to control but it was tough.
A: It was not a good idea to listen to the touching words because –  You would have wasted the opportunity to Nianfo, and  as you mentioned, it stirred your emotions till there was tearing, which is directly against the Second Great Essential taught by Great Master Yinguang: https://purelanders.com/2015/12/30/the-second-great-essential-%e7%ac%ac%e4%ba%8c%e5%a4%a7%e8%a6%81-pure-land-passport-section-3c
Such tearing along with the others can in turn stir much attachment in the deceased, thus disturbing his peace of mind, which endangers the quality of the last thought before rebirth, which might transpire at any moment. If you focus on sincere Nianfo only, there will be no extra ‘space’ in your mind at the same time to be emotionally disturbed. You should see yourself as offering possibly the last sliver of true hope of spiritual rescue, among all present.
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How To Help (Dying And) Deceased Non-Buddhists