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Regarding the confession, I've been doing some studying on the Baptist Faith and Message and it appears to have come from the New Hampshire confession. I don't know what came before that one.
James Boyce, a Baptist with reformed leanings and former Professor of Systematic and Polemic in Theology at The Southern Baptist Theological Seminary once (1887) stated,

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The Lord's Supper is an ordinance of Jesus Christ, to be administered with the elements of bread and wine, and to be observed by his churches till the end of the world. It is in no sense a sacrifice, but is designed to commemorate his death, to confirm the faith and other graces of Christians, and to be a bond, pledge and renewal of their communion with him, and of their church fellowship.

J.L. Dagg, stated,

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The Lord’s Supper was designed to be a memorial of Christ, a representation that the communicant receives spiritual nourishment from Him, and token of fellowship among the communicants.
The Abstract of Principles used today by some Baptists that have reformed leanings state,

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The Lord's Supper is an ordinance of Jesus Christ, to be administered with the elements of bread and wine, and to be observed by his churches till the end of the world. It is in no sense a sacrifice, but is designed to commemorate his death, to confirm the faith and other graces of Christians, and to be a bond, pledge and renewal of their communion with him, and of their church fellowship.
This view (Symbolic Commemoration or Memorialism) was popularized by Zwingli. Moreover, this viewpoint is often historically associated with the Anabaptists of the Radical Reformation (i.e., Conrad Grebel (co-founder of the Anabaptist movement and the "Father of Anabaptists") and Felix Manz (co-founder of the original Swiss Brethren Anabaptist congregation and the first martyr of the Radical Reformation), and the English Baptists.

Thus, this view has been in Baptist circles for some time.

Reformed and Always Reforming,