Should a pregnancy test a week after bleeding be able to tell me if it was implantation or a period?

You raise more questions than answers. Implantation doesn't typically cause bleeding, but a miscarriage shortly after implantation is often the cause of a "heavy period" since it is often still on time and the products of conception are too small to appear as anything but blood. A positive pregnancy test in the context of any vaginal bleeding needs to be investigated to rule out ectopic, etc.

Conception occurs within day or so of the sex. It takes about 3 weeks for the embryo to make enough HCG for a positive pregnancy test. Blood test may be done a day after a missed period. Home pregnancy tests should be done 4-5 days after the missed period. If the result is negative repeat the test in 4-5 days. Use first morning urine and follow instructions for the test carefully.