I believe that eternal life begins the moment we are saved, having saving faith, which "comes by hearing, and hearing by the word of Christ." And as you have pointed out, eternal life is not just unending life, but abundant life. Jesus came that we might have this abundant life, which is His life in us.
Romans 10:13-17
“WHOEVER WILL CALL ON THE NAME OF THE LORD WILL BE SAVED.”
How then will they call on Him in whom they have not believed? How will they believe in Him whom they have not heard? And how will they hear without a preacher? How will they preach unless they are sent? Just as it is written, “HOW BEAUTIFUL ARE THE FEET OF THOSE WHO BRING GOOD NEWS OF GOOD THINGS!”
However, they did not all heed the good news; for Isaiah says, “LORD, WHO HAS BELIEVED OUR REPORT?”
So faith comes from hearing, and hearing by the word of Christ.
John 10:9-10
I am the door; if anyone enters through Me, he will be saved, and will go in and out and find pasture. The thief comes only to steal and kill and destroy; I came that they may have life, and have it abundantly.