Neil Armstrong [...] said:
I have ridden on 13 different rocket engines

I can't match that number:

by type:
X-1b
X-15
Titan-II first stage
Titan-II second stage
Agena (well, it was planned....)
S-IC
S-II
S-IVB (I'm being generous, both S-II and S-IVB used J-2s)
Apollo SPS
LM Descent
LM Assent
That's 11.

By number:
1 X-1b
1 X-15
2 Titan II first stage
2 Titan II second stage
5 F-1s
5 J-2s
1 J-2
1 SPS
1 LM Descent
1 LM Ascent
That sum is too high even counting just the moon flight.

Assuming Armstrong is correct [sic!], what am I missing?

What about that lunar lander trainer vehicle that almost killed him? I
believe that it had two different engine systems installed.