Most of the Chinese, Korean and Japanese people that I speak to say
that there are only two strokes in 子 and 辶, saying that nobody does it
using three strokes.
Well, one thing is for sure, all of the people you spoke to who attended school would have learned these characters with 3 strokes, because officially (education ministries, etc.) they have 3 strokes in Japanese Kaisho (楷書{かいしょ}), Chinese 楷書 (both simplified and traditional) as well as Korean 楷書 Hanja. Kaisho is the name of the type of calligraphy that has been standard for Chinese characters in public education for a long time.
Yet, on every single official source (Chinese government sources,
Japanese dictionaries, online dictionaries) it shows each element with
three strokes. If the people do it one way and the official sources do
it another way, what is the correct way to do it?
Scholars and the government promote and teach an official ideal ("correct") system of writing and the people use it in daily life. What is correct may depend on the circumstances and what is required.
For instance, if you answer that 子 has two strokes instead of three on an elementary school writing test in Korea, Japan, China, Taiwan, Singapore etc. your answer will be marked as incorrect by your teacher. However, if you personally write a letter to someone named Keiko (圭子) and you write the second character without lifting your pen, it's unlikely that she won't be able to read it.
In Japan, there is no official government-mandated stroke order for
each character, which means that the official stroke order is decided
by the collective will of the people. Does this even matter?
The Japanese government has published guidelines which are followed by almost every publisher and educational institution in the country and these include the number of strokes in each of the Joyo Kanji. I would say that whether it matters to a learner of Japanese probably depends on their goals in learning to write, just like it does for someone learning to write in English.