Nope. Pascha does not begin on Lazarus Saturday. Pascha begins at the beginning of the Liturgical day on Pascha day - which would be at the singing of the Great Prokeimenon "Arise, O God, and judge now the Earth..." The Prokeimenon is the liturgical moment (during vespers) when the next day begins...
You're right that this is technically the time when Pascha starts. My question to orthodoxos was meant to make a point. The exact time when Pascha begins is really an organic thing. I`ve heard it said that it is like bread dough rising. The Church is a vital, organic, living body where living Tradition makes its home, so this should not surprise us. Look at the texts of Holy Week. There are very strong resurrectional elements present already at the Matins of Great and Holy Saturday. And even at Nocturnes Pascha night, there are still texts that hearken back to the entombment and the crucifixion. (Another point that one could draw from this is that the crucifixion and the resurrection are always strongly linked.)
Your response to orthodoxos's question is a good one. I think one could also answer that the process is an organic one that doesn't have an exact beginning (when considering it simply in the temporal sense.) John the Forerunner is definitely an Old Testament figure. You could say that the Theotokos links the Old and New Covenants. I think that the New Covenant is not fully and completely inaugurated until the day of Pentecost.
BTW, are you sure you are right to call "Arise O God" a prokeimenon? It takes the place of the Alleluia verses at this liturgy. Also, it is not part of vespers, as the Divine Liturgy of St. Basil has already started at this point in the vesperal liturgy, so your (correct) statement that the next liturgical day always begins at the prokeimenon (of vespers) does not apply here.