Baptists, Reformed folks, Lutherans, Catholics and EOs all believe that baptism benefits somebody. But who does baptism benefit? For the sake of simplicity I'm going to collapse everything into the Baptist view, the Reformed view, and the Catholic view. EOs and Lutherans will have to forgive me, but you must admit that your view on baptism is very close to Rome, if not identical. Likewise Charismatics, anabaptists, and the like share the Baptist view.
Baptist View - Baptism is mainly for the benefit of the baptized believer. It should only be applied to those who profess faith in Christ. The church witnesses baptism but only like those who are witnesses in a wedding ceremony. They may celebrate with the person being baptized, but the baptism is not about them. The person being baptized is publicly professing their faith in Christ and experiencing a kind of bench mark in their Christian faith.
Catholic View - Baptism is for the benefit of the baptized person, whether they are adults or infants. This is because baptism regenerates the person being baptized, translating them from darkness to light, from death to life, from citizenship in the world to citizenship in heaven. In the ancient church sometimes baptism was not even celebrated in the presence of witnesses because the witnesses are welcome, but not necessary for the event.
Reformed View - Baptism is for the benefit of all believers. The Westminster Confession of faith says that a sacrament is a holy ordinance instituted by Christ, wherein, by sensible signs Christ and the benefits of the new covenant are represented, sealed, and applied to believers. Baptism only benefits believers, but every baptism that a believer witnesses is a benefit to them. Every time the believer sees a baptism, he is seeing a visual representation of the gospel and by faith he grasps Christ and the benefits of salvation. This is why it's essential that baptism is celebrated in public worship where all believers can benefit from it. The Reformed baptize children simply because we are commanded to do so. Adult believers benefit from seeing the baptism of children. The children themselves do not benefit from their baptism until they are old enough to believe the gospel (whenever that may be).
The Reformed view is the most difficult one to grasp, but I believe it is the most biblical and the correct view. I imagine, however, that it will be the minority view on these forums.