So he introduced the senate version and voted against it?
This will take a minute to explain, as it requires delving into how Congress works. I'm including links to all of the congressional actions on the matter. They help.
1) Cruz introduces the Senate version of the bill (
https://www.congress.gov/bill/117th-congress/senate-bill/705) in March of 2001, three days after the House version (
https://www.congress.gov/bill/117th-congress/house-bill/1608) is introduced. Note that Cruz had already introduced this in the preceding Congress without result.
2) Both his version and the House version end up sitting dead in committee (see links).
3) The big spending bill originates in the House (
https://www.congress.gov/bill/117th-congress/house-bill/2471), as it has to. This starts as a more limited bill.
4) This hits the floor and ultimately ends up being the piece of legislation that becomes the omnibus bill, via this vote which also provides for voting on establishing the January 6 committee (
https://clerk.house.gov/Votes/2021190). In the process, Ports-to-Plains gets rolled in from the transportation bill.
5) This ends up in the Senate, where it passes by voice vote with an amendment related to Haiti (
https://www.congress.gov/bill/117th-congress/house-bill/2471/actions - fourth item from the bottom). This keeps him (or anyone) from having to go on record.
6) The House marks this up out of conference (
https://clerk.house.gov/Votes/202266) and sends it back to the Senate.
7) Cruz votes against the final version of the spending bill (
https://www.senate.gov/legislative/LIS/roll_call_votes/vote1172/vote_117_2_00078.htm), but it passes
8) Biden signs it into law
In general, most members in the minority/opposition these days will refuse to vote for an omnibus spending bill or a continuing funding resolution. This is so they can paint the other side as big-spending wastrels with the wrong priorities in the next election. If some people need to cross the aisle to keep the government open, usually that's some combination of the more moderate members looking for a bipartisan reputation, leadership or the safest members of the minority. That latter group either don't fear a challenge from either side, or just won re-election. You can see that pattern in the final Senate vote in step 7.
Another way to say that is: once Cruz gets the amendment attached he knows he's golden. The Democrats have to pass the omnibus, and it's not like he was trying to pass a poison pill to kill it with Ports-to-Plains. He was then able to free-ride on their votes when it came time to pass the omnibus, because it's Thune's job to know whether or not the votes exist to pass it with or without Cruz on board.
What the White House said is deeply misleading about how the process works, in the sense that members of both parties do this all the time. Cruz is an easy target and the White House despises him, so the White House said something that is substantively accurate while neglecting to tell the whole story. It's worth noting that the Republican sponsor of the corresponding House legislation also voted against the omnibus, but the White House didn't call him out about it because he doesn't have the target on him that Cruz painted on himself.
That story doesn't fit well in a thirty second news clip, and any attempt to explain how the process works is virtually guaranteed to make Cruz look bad, so the White House engaged in a little political theater to make Cruz squirm.