Why do I get different results when I use the pictured 10 mfd capacitors?
Using the same probe start ballast (pictured) and a new 175watt MH Standard (brand) lamp:

1 - Good run, excellent spectrum, sodium driven to self-absorption, scandium lines prominent.
2 - Poor light, mercury lines strong, but sodium 589 doublet and scandium lines just visible.
3 - Lamp does not ignite.
4 - Takes a long time to run up, but the sodium 589 is not broadened or self absorbed. Under-driven.
5 - Fires up, mercury lines only, then fails after a few minutes.

Why are the capacitors so different?
What is a "protected" and "not protected" capacitor?

It's likely that 4 of the 5 capacitors are failing, #2 looks pretty old and the Pulsrite one (#3) seems to fail a lot from what i've seen.

If you have a capacitor meter (or a multimeter with a capacitor setting), you might be able to test if any of them meet or are within tolerance of their 10uf rating.

It's possible, but 4 out of 5 capacitors made for HID lighting, the odds are not right.

I neglected to mention that I used more capacitors, particularly three different Aerovox #5's, all these acting the same: igniting almost normally then thinning of the arc until it can't be sustained. Something else is happening like some sort of compatibility.

There your problems Those Capacitors i would throw those away.  I would use metal can Motor run cap
such as Am Rad 10uF  440v caps.  they hold up in
A/c unit very well. also  There Anerican made too. I  use them for my street lightsit very important to use quailty caps.  you can  go to any electrical supply house and buy motor run caps.. I get mine from Scrapped out a/c units so it dont cost me a dime.  you can go online even
buy caps from home depot.  if your lighting cap such as
odd ball value such as 26 uF.   I use more common 25 uF.  1 uF not going make much difference in larger size..

It is not uncommon to find "all out of all" capacitors in a lighting installation failed within few years

Yes that true Esp for junk caps like these. first one on left that metal can Areovox is ok cap.

but When cap manufacfure Am rad take so much pride
and they give 5 year warrenty on them. that tells ya somthing there quality in these.  I have seen GE cap blow lids off.  the advance plastic cap lose uf.  Ge cap
is so bad in street lighting and HVAC industry killed so many  ballast and motors due poor quality. it actually worse than china made ones. so when i see those plastic or GE caps i usually ditch those and replace them for better caps.  those MV in parking lot have westy life gaurds and Am rad caps.  it still going to this day and none out! after in years of service.