There are some misconceptions in your question.
1) USB wall adapter and battery charger are the same thing?
No, they are different circuits. The AC-to-USB adapter for a phone or similar device is the equivalent of the wall adapter to a laptop, it provides a stable voltage from which everything inside the device runs. The voltage output will be 5.0VDC and current capability may range something like 0.5A to 2.0A depending on the model. The official USB 2.0 spec a few years ago was capped at 0.5A per port, but Apple made their own customized version in order to provide more power to charge the iPad faster. Many third party vendors reverse engineered and copied the design modification, so 2A USB adapters aren't unusual now. A new official USB 3.1 spec ranging up to 20V x 5A = 100W is on the horizon for future devices. See Wikipedia on USB Power.
Meanwhile, the actual battery charger circuit is inside the device. It takes the USB adapter output as its input and regulates it (again) to what the battery should have as a charging input. For a lithium polymer battery the charger limits both the voltage and current into the battery, with voltage limit set to something like 4.0 to 4.2V and the current limit to a 1C rate at most, for a 1 hour charge. Likely somewhat slower in order to do as little damage to the battery as possible while giving the user an acceptably fast charge time. Some algorithms go quickly for the first 80% full and slower for the last 20% full.
2) The USB adapter sets the charge rate of the battery?
The USB adapter doesn't say to the device "charge at this rate," it says "this is how much power I can put out" and the device adjusts itself accordingly. The maximum power level is supposed to be negotiated between device and host according to the USB spec, starting from a minimal level and granting the device more power upon request. For details see the USB Implementers Forum and this StackEE post. In practice, some (most?) hosts offer the maximum of which they're capable at all times the port is switched on. Apple's proprietary scheme involves pullup & pulldown resistors on the data lines to signal how much power is available. The upcoming USB 3.1 spec may handle things differently.
The USB adapter only affects the charge rate if it forces the charger to go slowly because it doesn't have enough power input to charge as fast as it would like. For example, an iPad Air ships with a 5V x 2.4A = 12W adpater, has a 32.4W-hr battery, and thus could not charge any faster than 32.4/12 = 2.7 hours or about a C/3 rate (not 3C rate). With efficiency losses, slower charge after 80%, etc., it's more like a 4 hour charge from 0% to 100%. Reference: AnandTech. If you plug the same iPad into a smaller iPhone USB adapter which can only output 5V x 0.5A = 2.5W, you'd necessarily see something like 5x longer charge time because the adapter is only putting out 1/5 the power. Presumably the results are similar for other combinations of device and port, BUT with a badly designed device &/or adapter it could be the case that a heavy load simply overloads the adapter and trips a fuse, or worse.
3) Device makers charge at 3C rate or "don't care" about damage to battery?
Reputable device makers care very much about damage to the battery. A badly designed product can literally catch on fire. There may be a big difference in quality or reputation management between a large public company and a no-name brand from a distant corner of the world. Nonetheless it's not typical to charge at 3C except in devices specifically designed for it (using a high-rate battery) such as cordless drill. The chart you reference was probably created in a test lab where the batteries were charged and discharged directly on battery testing equipment set to harsh parameters to see what would happen, not in a consumer device.
4) Charger ICs don't allow setting maximum charge current?
The ICs do allow such setting. They have to permit the charge current to be set somehow by the device designer in order for the chip to be used in a broad range of devices. Have a look at the data sheets for chips from Texas Instruments, Maxim, Linear Tech. In a phone this function may be integrated into the power management IC rather than a standalone charger IC. Again, this is handled by the device, not the wall adapter.