technically, when you put in your two week notice, you have already quit, the two weeks is for the benefit of the company. It is to the discretion of the company to decide how best to utilize you during those final two weeks, if at all. Removing you from the regular schedule is common practice. I would never put in my notice until after I found a new job. Most places understand that, and ask when you will be available if hired. That's when you say, "I have to give my two weeks notice, but will be available directly after that."
As you have found, two weeks doesn't mean two weeks to employers. Every time I put in my two weeks notice, I was out of work the following day.
"I would rather be exposed to the inconveniences attending too much liberty than to those attending too small a degree of it." Thomas Jefferson