There is no difference between the day of Christ and the day of the Lord. Other translations even have "the day of the Lord" there instead of "the day of Christ". There is no basis whatsoever for making this distinction. Paul was talking about the day that Christ returns and we're gathered to Him and he was calling that the day of the Lord (or the day of Christ, if you prefer - same thing).
2 Thessalonians 2:1 Concerning the coming of our Lord Jesus Christ and our being gathered to him, we ask you, brothers and sisters, 2 not to become easily unsettled or alarmed by the teaching allegedly from us—whether by a prophecy or by word of mouth or by letter—asserting that the day of the Lord has already come. 3 Don’t let anyone deceive you in any way, for that day will not come until the rebellion occurs and the man of lawlessness is revealed, the man doomed to destruction.
First, Paul referenced "the coming of our Lord Jesus Christ and our being gathered to him". Then, instead of saying all of that again, he simply referred to the day of His coming and our being gathered to Him as "the day of the Lord" (or "the day of Christ"). And then in verse 3 he referred to it as "that day". The context is regarding the day Christ returns and we're gathered to Him. That day when Christ returns and we are gathered to Him will not occur until the rebellion (falling away) occurs and the man of lawlessness (man of sin) is revealed first. That is clearly what Paul was saying in this passage.
In saying "that day", this references "the day of Christ/Day of the LORD" only, it does not reach back to reference "the coming of the Lord, and by our gathering to Him". The real question here is whether or not the "day of Christ" or "Day of the Lord" is idiom for our being gathered to Him.
It's not a translation issue, it's a manuscript variation, two completely different words are used in this place in the different manuscripts.
Philippians 1:6 KJV
Being confident of this very thing, that he which hath begun a good work in you will perform it until the day of Jesus Christ:
Another question is, should "day of Christ" be understood to be the same as the "day of Jesus Christ."
Something I've seen over the years, in these places where sometimes He's called Jesus, and sometimes Jesus Christ, and sometimes Christ, and sometimes the Lord Jesus Christ, I think these are all used with distinct reason. For instance,
Romans 6:8-11 KJV
8) Now if we be dead with Christ, we believe that we shall also live with him:
9) Knowing that Christ being raised from the dead dieth no more; death hath no more dominion over him.
10) For in that he died, he died unto sin once: but in that he liveth, he liveth unto God.
11) Likewise reckon ye also yourselves to be dead indeed unto sin, but alive unto God through Jesus Christ our Lord.
Vs. 8 & 9, we be dead with Christ, Christ being raised from the dead, "Christ" speaks to Jesus' annointing to be our Savior. So we are dead with Christ, He rose from the dead, these are the mechanics of our salvation, if you will.
Vs. 11, we are alive unto God through Jesus (the Man Who is our intercessor) Christ (our annointed Savior) our Lord (whom we obey if we are true).
The say the proof of the pudding is in the eating, the proof of our believing is in the doing.
Much love!