The two statements in quotes are surely different. But the first can also express a moral standpoint: that it is morally important to achieve a low murder rate. This moral standpoint is reflected in various more specific claims.

1. We should not inflict punishment or pain on anyone unless doing so produces some good for others (e.g., by preventing the person from offending again or by deterring others).

2. We should inflict pain whenever doing so produces some greater good for others.

This second claim is highly problematic insofar as it may justify "punishing" the innocent when doing so helps deter real criminals. For this reason, those who hold deterrence to be morally important often claim instead:

3. In deciding how severely to punish specific types of crime, we should take into account how much of an impact greater severity would have on the frequency of this crime.

This third claim is consistent with the idea that people may be punished only for having done something wrong. But it allows two kinds of conduct that are equally wrong to be punished differently. One crime is punished lightly, yet some equally bad crime is punished more severely because here such punishment is more effective in getting the crime rate down. Moral arguments are made both for and against this claim.

It's apparently really expensive to shepherd a capital case all the way through to eventual execution--some say, much more expensive than life in prison.

http://www.deathpenaltyinfo.org/article.php?did=108&scid=7

That aside, there are plenty of other ethically relevant considerations. You can free a prisoner if the conviction is overturned, but you can't bring him back to life. Few of us, when actually faced with a choice, would choose death over life in prison. Nasty as it is, prison is not necessarily intolerable, and it may allow for the pursuit of important aspects of a good life. And even if capital punishment is already used for some particularly horrible crimes, expanding it to others may face objections of the sort people raise against capital punishment in the first place, to do with fairness of application, the value of life, the inhumanity of easily avoidable killing, and the merits of mercy. Even if these objections did not convince you to keep the mass murderer alive, they might move you with respect to the enraged spouse or the drug dealer or the white-collar felon. They might not, certainly, but they need at least to be considered before you can reasonably judge that it's at least as good to kill these people as to lock them up.

Any realistic criminal justice system will make both types of error: T-errors (terrible future crime committed by one wrongly acquitted) and I-errors (innocent person wrongly convicted).

It seems morally more appropriate here to compare alternative systems feasible for the same society in terms of the total number of errors, not in terms of the ratios you focus on (ratio of T-errors to total number of acquitted, ratio of I-errors to total number of convicted). To see this, take a criminal justice system under which 100,000 are acquitted of whom 2000 are wrongly acquitted and go on to commit a terrible crime. Now suppose we modify this system so that many more innocent people are tried and properly acquitted. So now we have 200,000 acquitted of whom 2000 are wrongly acquitted and go on to commit a terrible crime. So we got the T-error rate from 2 percent down to 1 percent. But have we improved the system? Surely not.

So I rephrase your question this way: In designing our criminal justice system, should we give greater weight to avoiding T-errors or to avoiding I-errors?

There is a widely held view bearing on this question. In one famous formulation by William Blackstone (www.law.ucla.edu/volokh/guilty.htm), it says: “It is better that ten guilty persons escape than one innocent suffer.” I think what stands behind this view is the idea that we as a society bear a far weightier responsibility for harms to innocents that we directly bring about than for harms to innocents that we merely might have prevented.

This idea is familiar in pedestrian contexts. It would be wrong to kill one innocent person in order to save another. It would be wrong to harm one innocent person in order to rescue another from a similar harm. It would be wrong to risk the life of one innocent person to prevent a risk to the life of another (risks being equal). And it would be wrong to cause a risk of harm to one innocent person in order to prevent the same risk of harm to another. In all these cases, agents ought to give more weight to harms and risks of harm they directly bring about than to similar harms or risks of harm that they merely fail to prevent.

This almost answers the reformulated question: We should give greater weight to avoiding I-errors than to avoiding T-errors. I say "almost", because the average harm associated with errors of each kind may not be equal. So the conclusion might conceivably be overturned if the harm done by the average I-error is just a few days in jail without public stigma while the average harm resulting from the average T-error is a minor massacre. But this is not true in the real world. Here, in order seriously to reduce the number of terrible crimes committed by persons wrongly acquitted, we would have to make it easier to convict people accused of very serious crimes and then take those convicted out of circulation for very long periods or forever. And doing this will inevitably increase the number of innocent people who, wrongly convicted, are locked up for very long periods or even executed.

I agree that your solution works as well or better. Here are two different arguments a consequentialist might make.

(1) Suppose all attempted murders are punished equally, regardless of success, with each attempt being punished with 6 years in jail and 30% of punished attempts successful. Now consider this reform: We increase punishment for successful attempts from 6 to 13 years and decrease punishments for unsuccessful attempts from 6 to 3 years. This reform leaves constant the jail time per punished attempt (which consequentialists typically count as a negative): 13 x 30% + 3 x 70% = 6. (Obviously, the numbers here are just for illustration.) But the reform is likely to increase deterrence, because prospective murderers are going to focus more on the "successful" outcome that risks a 13 year penalty than on the (actually more likely) "unsuccessful" outcome that risks a 3 year penalty. As a result of better deterrence, fewer murder attempts are made, fewer people are murdered, and fewer years are spent in jail. (Obviously, for this argument to work, I must be right about the psychology of prospective murderers.)

(2) Whether an attempted murder succeeds or not is sometimes due to luck. But it is often correlated with features of the agent (greater relevant abilities, will power, nerve, lesser moral scruples, and so on). Therefore, those whose murder attempt succeeds tend to be more dangerous, on average, than those whose murder attempts fail. Also, given that criminals often commit further crimes, an extra year in jail for a more dangerous criminal better protects the public from repeat offenses than an extra year in jail for a less dangerous offender. It therefore makes sense to shift jail time from unsuccessful to successful attempters.

..... But, if these are good arguments, might consequentialists want to advocate that only successful attempts should be punished?

On the former question, how much I pay for car insurance is a function of how much accident claims cost insurance companies. People who do not wear seatbelts cause themselves greater injuiries, which lead to higher insurance claims, which lead to higher rates for me. So it's far from clear that one who doesn't wear a seatbeltharms only h'erself (even waiving the harm s'he might be doing to family and friends). The same is true in the latter case: It's not entirely obvious that people who abuse cocaine, say, harm only themselves. More needs to be said, obviously.

There is at least one good consequentialist reason for punishing attempted murder less severely than murder. If the two crimes are punished equally, then the law will not deter someone who has tried and failed to murder from trying again!

To the extent that the "successful murderer" is simply luckier than the "failed murderer," your question raises the vexing problem of moral luck. Consider the following two cases:

1. John drives after drinking way too much at lunch and is pulled over almost immediately and arrested.

2. Jack drives after drinking way too much at lunch and almost immediately runs over and kills a family of four crossing the street.

It seems that there is no morally relevant difference between these two cases. That is, the only difference between them seems to be that John was lucky enough to have been pulled over before he could cause any serious harm. Yet (at least in our current criminal justice system), we punish Jack much more severely than we do John. John is charged with driving under the influence, but Jack is guilty of vehicular manslaughter. Should John receive more lenient treatment merely because he was lucky?

Two of the best-known essays on moral luck are by Bernard Williams and Thomas Nagel. You can find them here and here, respectively. There's also a nice (and free) overview in the online Stanford Encyclopedia of Philosophy.