iPhone 4, when on, only charges hooked up to Mac, not any other USB source

I have recently bought a refurbished iPhone 4, to use with a Cycling App (cyclemeter). It is working fine, except when it comes to charging.

When I have the iPhone on and running any app, the only time it will indicate it is charging, and increases the battery percentage, is when I have it plugged into my Macbook 2.1.

When connected to the USB wall charger that came with it (probably not original Apple charger), it does not show it is charging, and the percentage drops. For the bicycle use, I have a USB charger that runs off a 11.1V 2200 mAh battery, so I want to keep recharging the iPhone during long rides. But it does not indicate it is charging either, when plugged in, and the percentage drops.

However, if I turn the iPhone off, and leave it alone, later when I turn it back on, it has been charged by the wall charger or 11.1V battery powered USB charger.

So, it is charging, but not the way I need for it to do. I looked at some Apps, and have tried Battery Doctor. But the charge status is the same - when turned on, the iPhone does not charge to increase the percentage when connected to the wall charger, or connecter to the 11.1V battery powered USB source. It only charges when connected to the MacBook.

Note - I am using the same charging cord for all, the only change is what USB source it is plugged into.

Are there any suggestions? If a rechargeable Battery Bank would do it, I'd go for that, but I'd need to know of some models known to work with an iPhone 4, as I am sure that if some work, a lot won't.

Could it be firmware that I need to reset or something like that?

BTW - It is running IOS 7.1.2