Your question is difficult to answer due to the way it's worded.
First, what does "Western Tradition" mean? The Latin side of the church (Carthage, Rome, Gaul, etc.) held to the same Tetraeuangelion as the Greek and Syriac church did.
Second, When Luke mentions that there were others before him who "stacked up an orderly account" (“ἀνατάξασθαι διήγησιν” (Λουκᾶν 1·1 THGNT-T)), he is not speaking about his fellow gospel authors. He's just mentioning the fact that there were other accounts that people had written down. His account was different than these early (non-canonical) accounts, however, for two reasons:
- He made use of direct witnesses and servants of the word (“αὐτόπται καὶ ὑπηρέται” (Λουκᾶν 1·2 THGNT-T))
- His account was recorded "accurately and orderly" (“ἀκριβῶς καθεξῆς σοι γράψαι” (Λουκᾶν 1·3 THGNT-T))
He is definitely and obviously not speaking about John's gospel, since John hadn't written it yet.
Third, There's absolutely no evidence from you that the synoptic gospel writers have a different reckoning of time for Jesus' ministry than John's gospel. The sheer fact that the synoptic gospels record events both from Jesus' early ministry and from Holy Week dispels your statement.
Finally, when speaking of evidence of a canon, we can look in (at least) two directions:
Manuscripts: The manuscripts handed down to us all contain all four gospels without a hint of any ordering one in priority before another. Examples:
In the West, one of the earliest manuscripts of the Tetraeuangelion is Codex Vercelli. It contains all four gospels recorded in order of length. So it's recorded and bound in this order: 1) Matthew; 2) John; 3) Luke; 4) Mark.
In the Greek-speaking areas, take, for example, The Chester-Beatty Papyrus (P45-P46); contains all four gospels.
In the Syriac-speaking areas, take, for example the Curetonian Syriac. It contains all four gospels of the Tetraeuangelion.
Furthermore, not only do they contain the four gospel collection (Tetraeuangelion); they also add this preface to them all:
- ⲉⲩⲁⲛⲅⲅⲉⲗⲓⲟⲛⲕⲁⲧⲁ...
- Euangelium Secundum ...
- ”ܕܐܘܢܓܠܝܘܢ ܕܡ̈ܦܪܫܐ ܡܬܝ“ (Matthew 1:0 SYRIAC-C)
The gospel "according to [gospel writer]." Notice, that there is nowhere recorded in any manuscript any hint that any of the gospel writers was more or less in importance than the others. Also, note that never, in any manuscript, do we have the superscript, "Luke's gospel." The possessive is never added. The gospels do not come from Matthew, Mark, Luke, John. And they do not belong to them. They belong to Jesus. Instead, they are the gospels "in line with" or "according to."
Church Fathers: One of the patterns we find very early in the church is the Church receiving and treating the Tetraeuangelion as a unit/collection of four equally-weighted gospels is how the early church fathers speak of them. As early as the time of Irenaeus (130–202), the Fathers begin to speak of the four gospel writers in terms of four beasts.. But how they speak of them is the important issue. They do so as an answer to the question, "why four gospels?" In their answers all four are canonical.
As a summary, when you assert claims like you have here, you need to provide evidence. Also, it would be wise to break down your questions into separate questions. Examples:
- Does Luke explicitly say his gospel was the right/only one?
- Was Lazarus a source for the gospel according to John?
- Do the gospel writers have a discrepancy in the years recorded?
- Did any of the gospel writers assert that 'their' gospel was before or after (in priority) the other gospel writers?
If you don't, at the very least, your question becomes an incoherent mess. But, more likely, your question will either not be answered at all, or be shredded due to how it is framed (it seems to be driving an agenda rather than asking for knowledge).
Recommendation: If you are at all interested in learning more about the NT canon I suggest The Canon of the New Testament: Its Origin, Development, and Significance. It is very well-researched. And it would quickly clear up the muddiness of your questions.