TL;DR: No, Intel did not want x86-64, they had IA-64
The rest was FUD, power play by Microsoft and market reality.
Zac67's answer may point out some important confusion, except in my recollection there was a bit more to the story than confusion and Intel creating quite some hot air about countering AMD64 with their own design. Looking back, I'm not sure how much of that was true, but in the industry it was widely seen as such, and so did I. To understand it may help to take a look at the bigger picture:
Some History
Intel had done a very similar move just the year before by introducing SSE with the (Pentium III, 1999) to counter AMD's 3dNow! (K6-2, 1998), thus when AMD released their AMD64 (aka x86-64) spec in 2000, everyone was expecting intel to show off their own thing, ofc, not compatible to AMD, as with SSE before.
Previous extensions were usually only announced shortly before fitting Hardware (CPUs) were introduced. In contrast AMD64 was announced in 1999 and publicly documented in full in 2000, while first hardware (K8) took three more years until 2003 as Opteron in Spring and Athlon 64 right for Christmas 2003.
The Early Naughties (00..05)
It further helps to remember the over all time line, as Intel entered naughties shell shocked by AMD's superior 1999 K7/Athlon CPU line that clock for clock outperformed the Pentium III and reached higher clock rates at the same time. The Pentium III was still based on a 1995 Pentium Pro (P6) architecture. It wasn't until the 2007 core architecture that Intel regained ground with the Core design, the first really new architecture - no, the Pentium 4 was an utterly failure. Great on first sight, but deadly crippled when running real world loads. Also still a direct P6 offspring (thus named P68).
Enter Microsoft
Now add another major player to this 2000 to 2005 backdrop: Microsoft
It was the time that pushed very hard into the server market. They had just released Windows 2000 Server in 1999 and started to work on Windows Server 2003, intended to be a distinct, server first product (hence the name change). Windows 2003 was also posed to support IA-64, Intel's sole 64 bit product in form of the Itanium CPU. Adding AMD64 support was seen as a possible additional target, especially as it supports upward compatibility with acceptable performance - frankly, Intels IA-32 support on IA-64 sucked.
But MS did neither have capacity nor any intention to support two different 64 bit extensions for x86 at kernel level - which is the really important part here. Having a bunch of additional instructions at user level (think MMX or SSE) doesn't matter at all, while having more registers to save and restore are minor details usually covered by the most basic HAL-functions. But AMD64 not only introduced new 64 registers and addressing, but also a new memory management model, which at least requires adapting and hardening all modules touching virtual memory - which can be a lot. This is where the cost of two different 64 bit extensions for MS were and why they wanted to avoid them as much as possible.
Intel Dragging its Feet
While Intel was in 2000 still licking the Athlon wounds and pressing ahead with a firm commitment to IA-64, AMD delivered full AMD64 documentation in early 2000 and prototype hardware during 2002 to Microsoft. Everything a software developer could wish for. At that point Intel might have still convinced themself that it's a fast fading buzz. That is until AMD presented the Opteron 64 in April 2003 - right the same time as Microsoft premiered its Windows Server 2003. Of course shown off running at Opteron based servers - also the reason why the Opteron 64 was pulled before the Athlon 64.
Intel Being Dragged
And guess what? It was a success. Server operators saw the Opteron as a great way upward at a lower cost than IA-64. After all, they were faster and more cost efficient even without running a 64 bit OS.
So Intel seeing their customers going for faster and larger AMD64 servers instead of IA-64 (or more Pentium 4), needed a stop gap measure for their server business (Xeon). So a Pentium 4 variant supporting EMT64 was introduced a good year later in June 2004. (I believe there was also a Pentium 4 version shown at the Microsoft Server 2003 event capable of doing Intel64, but I can't find any evidence)
In fact, the exact timeline of AMD64 introduction with the Prescott line paints a good picture of that 'dragging'.
- In February 2004 Intel introduced a Pentium 4 using the 90 nm Prescott core as Family 15 Model 3. This core did already was already fitted with Intel64, but disabled.
- In June 2004 the Nocona based Xeon was offered with Intel64.
- Finally in November 2004 the Prescott E0 Stepping was announced which now also enabled Intel64.
Part of this forth and back can be heared in David Cutlers 2023 recollection recorded 2023 by Dave's Garage, which Sebastian Koppehelt found.
Fudding Names
Originally Intel named the 'new' extension EM64T for Extended Memory 64 Technology, which nicely shows the whole reasoning behind adding it: the new Memory Management fitting what Microsoft implemented in Windows Server 2003 and the later High end Workstation Version Windows XP Professional x64 Edition (Not to be confused with the Itanium Version Windows XP 64-Bit Edition).
Now when those CPU finally arrived, Intel choose Intel64 as marketing name - obviously to counter AMD64 being mainly used. Over time public simply choose to call it x64 or likewise - today morphed into x86-64.
Conclusion
Intel had in 2000 no intentions to build a 64 bit x86. Their server focus (and that's what 64 bit was seen for) was at 100% IA-64 aka Itanium. They created a notable level of FUD - and maybe some half founded development as support - to stop AMD from gaining ground in server applications with their AMD64. Mostly felling how inefficient and dead end a 64 bit rucksack for a low end technology like IA-32 is.
Microsoft in turn was as well geared to IA-64 support in addition to 32 Bit x86 (IA-32). Supporting a 64 Bit x86 as well was fine to them. Except there was no time nor willingness to support two different 64 bit extensions. AMD delivered documentation early on and customer ready hardware right in time (2003) for Windows Server 2003 to shine. Intel could only react to jump onto the already rolling train. No room for any development of their own.