training while sick

I think it's a personal choice.  I like to train through it if I can.  However, I listen to my body VERY carefully.  If I don't have a fever, I'm more likely to train through it.  If I'm nausious, I tend NOT to train.

I tend to decrease the intensity first, then shorten or eliminate a workout if I feel too tired.  I don't want to push myself too hard and prolong the illness.

I work on getting even more rest, and drinking more fluids...

Rest.

Honestly, if it's merely a cold and completely above the neck, you're probably good to go. If it's in your chest or worse than just a little cold, I would rest. Your body needs the rest, and it's also not going to be able to recover as well from your training anyway if it's fighting something else. If you have a fever, definitely don't train.

x 2

If you feel that it's interfering with your ability to perform or recover from your workout, then I would just rest.  If you can't go hard during a hard workout, there's very little point to doing the workout and you'll just end up prolonging your recovery.

Resting for a day or two will not harm your performance in any significant way.

Cold, headache, slight stomach upset = train.  Anything else, eg. chest infection, very bad coughs = rest.  That's my general rule, although you've just got to listen to your body.