I believe that the present dispensation began when Paul started preaching the gospel of grace. Here are three quotes from him where he speaks of a "dispensation" that has been committed or given to him:
"If ye have heard of the dispensation of the grace of God which is given me toward you" (Eph. 3:2).
"Whereof I am made a minister, according to the dispensation of God which is given to me for you, to fulfil the word of God" (Col.1:25).
"...a dispensation of the gospel is committed unto me" (1 Cor.9:17).
The "dispensation" which was committed to Paul is in regard to "God's grace", a "ministry", and a "gospel." Here Paul sums up his dispensational responsibility:
"But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God" (Acts 20: 24).
I believe that Paul first began to preach the gospel of grace when he began preaching a gospel to the Gentiles at Acts 13:46-48.
The Bibical dispensation is a "stewardship" and all those who have been given a stewardship have a responsibility under that stewardship. What stewardship responsibility was exercised when you say that the present stewardship had its beginning?