Here is a full harmonization of all four of the gospel accounts, taken from the work of Wilbur N. Pickering. I cannot improve on his harmonization, so I am copying and pasting it. Your assumption that the women saw Jesus when running back to the tomb is not stated in the text. Although it is true that not all the gospels state specifically that Mary Magdalene was the first to see Jesus, there is no statement in any of them that she was not, neither is there a clear statement which imples that she could not have been the first to see Jesus.
THE RESURRECTION ACCOUNTS—A Harmonization1
Wilbur N. Pickering, ThM PhD
A rough sequence within the parallel accounts
Mark 16:1-3 // Luke 24:1;
Matthew 28:5-8 // Mark 16:4-8 // Luke 24:2-8;
Mark 16:9 // John 20:11-18;
Luke 24:36-43 // John 20:19-31.
The presumed sequence of events:
0. [Saturday—guards seal the stone and set up a watch (Matthew 27:62-66).]
1. Jesus rises from the dead.2
2. Early Sunday morning the women set out for the tomb—Magdalene (John.20:1); Magdalene and Mary (Matthew 28:1); Magdalene, Mary and Salome (Mark 16:1-2); Magdalene, Mary, Joanna and others (Luke 23:55-24:1, 10).3
3. On the way they worry about the stone (Mark 16:3).
4. Before they arrive an angel rolls back the stone, complete with earthquake, etc. (Matthew 28:2-4).4
5. They arrive and see that the stone has been rolled back, but the angel was no longer visible outside (Mark 16:4, Luke 24:2, John 20:1).5
6. Magdalene takes off immediately to tell Peter—Peter and John run to the tomb to see (John 20:2-3).6
7. Before Peter and John get there the other women enter the tomb, and see and hear the angels (Luke 24:3-8, Mark 16:5-7, Matthew 28:5-7).7
8. They leave the tomb in fear, saying nothing to the guards or anyone they chance to meet (Mark 16:8, Matthew 28:8a).
9. Probably right after the women leave, and before Peter and John arrive, the guards take off (Matthew 28:11-15).
10. Peter and John come and go [to their own homes] (John 20:4-10; cf. Luke 24:12 that is an historical aside).1
11. Magdalene returns to the sepulcher but doesn’t get there until everyone is gone (that’s why she thought Jesus was the gardener); Jesus appears to her first (Mark 19:9, John 20:11-17).
12. Then Jesus appears to the other women and they go on their way to tell the disciples (Matthew 28:9-10, Luke 24:9-11).3
13. Magdalene goes and tells the disciples (Mark 16:10-11, John 20:18).
14. Later in the day Jesus appears to Peter (cf. Luke 24:34).4
15. The Emmaus road episode (Luke 24:13-35, Mark 16:12-13).
16. Jesus appears to the eleven, Thomas being absent (Luke 24:36-48, Mark 16:14-18, John 20:19-23).
17. After Jesus leaves, Thomas comes in and they tell him (John 20:24-25).
Post resurrection day events
1. The next Sunday Jesus appears to them again and deals with Thomas (John 20:26-29).
2. Jesus appears to the seven beside the Sea of Galilee (John 21:1-22).
3. On a mountain in Galilee (Matthew 28:16-20).
4. Jesus appears to over 500, also to James (1 Corinthians 15:6-7).5
5. The ascension from Olivet (Mark 16:19-20, Luke 24:49-51, Acts 1:3-12).