Credit card size means it's scaled down by about 2? Difficult to read?
I have spent an estimated 30,000 km on the Thai railway network and don't remember having been asked for a passport! Most of these trips were staying in hotel, so I had my passport with me.
My experience:
6 months of carrying my passport with me every day and having it stolen once (Songkran 2013).
5 years of not carrying my passport with me and never been asked to see it on the street.
For this experience, I leave my passport at home unless I know I need it.
Every time I enter Thailand, I take pictures of front page, entry stamp, exit card with my phone, but do not make a photocopy.
However on a recent trip (Khon Kaen, Mahasarakam), I forgot my passport at home. Fortunately, an expired German ID card was accepted for check-in in both hotels I stayed (my valid German ID card was safe at home, like my passport). For that reason, I might switch to carrying a paper copy with me all the time. And should it be necessary, I rather give a paper copy to someone who asks for it, than my mobile phone. The downside is that I have to make a new copy every time I enter Thailand (which is 60 days or less on tourist visa).