The point being made in the article is that people find those believing less believable because of affirming and is suggesting that either all oaths should be the same or that they shouldn't affirm in public.
Yes, my mistake - I didn't read the full article.
I can certainly see that the type of oath/affirmation used might influence a jury member, whether consciously or sub-consciously, but I'm not sure you the claim that this adversely affects those who affirm is credible.
In our increasingly secular society, those who do not subscribe to a religious belief are becoming the norm, and I would suggest these people (including non religious not just atheists) are more likely to affirm. So the person giving a religious oath may be rarer and with juries also reflecting society (with perhaps 80-90% of people not having any meaningful involvement with any religion) these people may be seen as a touch odd and different, and that may work against, rather than for them.
But of course the bigger point is that the law of the land applies to all equally and hold no truck for religious belief or lack thereof, and so any oath/affirmation should be the same for all people, and would therefore have to be secular.
There can be no claim from a religious person that they cannot in conscience make the following statement:
"I do solemnly, sincerely and truly declare and affirm that the evidence I shall give shall be the truth the whole truth and nothing but the truth."
The reverse is not the case - no atheist can, in conscience make the following statement:
"I swear by almighty God that the evidence I shall give shall be the truth the whole truth and nothing but the truth."