I also think that subjective morality is absurd. ( by which I mean right and wrong are determined by either an individual or a specific society at a specific time). Just because there are differing believes about what constitutes right and wrong does not necessarily mean that morality is a man-made concept or relative. Belief does not always correlate to objectivity. For example, in the past many groups had differing views on the structure of the earth (some believed the Earth was flat, others believed the Earth was round etc), however this does not mean that the conception of Earth is subjective. The earth is objectively round. In a similar way, although societies and people at different times have had differing views on what constitutes right and wrong, it does not mean that right and wrong is subjective; one could just be wrong in his/her views about morality. Yes, it is true that people have differing moral beliefs, but that is not moral subjectivity. Moral subjectivity (or relativity) means to believe that right and wrong are itself dependent on those beliefs (which I think is absurd). e.g A moral subjectavist would claim something like "I think killing innocent children is OK, therefore it is OK" . An absolutist on the other hand, while recognizing the different moral beliefs, will always hold that there is moral system independent of individual beliefs.
The three major theories about morality (Deontology, Utilitarianism and Virtue Ethics) are all objective theories of morality. They all believe that certain acts are right/wrong independent of what human's think. Philosopher Kant for example saw moral laws as quite similar to the universal laws of nature (like gravity) and saw that it is only human beings who are intelligent enough to recognize and act upon these laws. Some theories say that good actions are those that promote the greatest amount of happiness (or well-being) for all sentient agents. Some theories say that good actions are those that conform to God's Will. Some theories say that good actions conform to Humanity's will. In all cases, the theories are objective.
OP is arguing that Moral Objectivity isn't based on what people think or do, not necessarily that it isn't based on people themselves. Many objective theories of morality are based upon people, but they are objective that they apply to all people. Other theories of morality even apply to all living beings! Killing may simply be wrong because it violates something valuable in itself (life) in an unjustified manner. What I think OP is trying to say is that just because people think something is right or wrong for them may not mean that action in actuality is right or wrong.