Difference between spec limits and control limits

To add to Don’s response, spec limits are not normally used on control charts.  They are used to determine your process capability and sigma value.  For example, if as you say, your spec limits are tighter than your control limits and your process is in control and producing consistent results, you may be producing defect-free products (within the spec limits).  On the other hand, you may be in control (within the control limits) but producing defects nonetheless if you are falling outside the spec limits. Hope this helps.

Your second statement is correct.  You want the control limits to be within the spec limits.