Its a shame that when speaking about Sound level meters some terms clash with each other. Weighting to me corresponds to the response time of the meter.
They don't really clash. In basic terms there are two seperate weighting protocols. The term weighting can apply to more than one measurement protocol.
One lives in the time domain (fast, slow, peak, impulse) and one lives in the frequency domain (A, B, C, D, Z).
In this technical sense, weighting means to give precedence to certain characteristics of the measurement be they in the time domain, frequency domain or both.
In the case of an SPL meter either one or both may be applied to the measurement depending on the required protocol and the application. Some but not all SPL meters may combine time domain weighting and frequency domain weighting setting in a single switch position if that is what the required measurement protocol calls for.
It is not unusual to see posts where a particular sound pressure is indicated by just XX dB. Without an indication of the weighing protocols applied to the measured value (time domain or/and frequency domain) the measurement is somewhat meaningless.
This is especially true when comparing measurement made be different people.
As you can see some people believe that the letter weighting applies to ballistics of the measurement, ( amplitude versus time-not correct) or that the term weighting only applies to one measurement protocol (again not correct).
I took just a quick look at the link that toddalin posted and it seems to give some good information.