Windows – I’m unable to boot the Windows 10 installer USB in UEFI mode

Squircle's answer and various comments are good; however, I want to address a few points....

I don't have the CSM mode option in my UEFI. Furthermore, I can't and don't want to convert my hard drive form MBR to GPT

It is possible to convert from MBR to GPT without losing data. My own GPT fdisk (gdisk) program can do this. I hear that some third-party Windows tools can do it, too, but I don't know the details. Microsoft's own tools cannot do this, though, so do not try to use them for the task. Note that MBR-to-GPT conversion is very simple compared to something like converting from one filesystem to another, so the risk is low -- but not zero. No operation that involves writing to data structures as sensitive as a partition table is zero-risk.

You say you "don't want" to convert to GPT, but you don't say why. If it's simply the hassle or risk of conversion, that's dwarfed by follow-on problems of the near-complete lack of support you'll face by using MBR. (I am, however, assuming that you have or can create adequate backups. If that's incorrect, FIX THAT PROBLEM IMMEDIATELY! Backups are NOT a luxury!) If, like some people, you're comfortable with MBR and just don't see a need to change to some [crotchety old-man voice]new-fangled thingamajig[/crotchty old-man voice], then I recommend you shed your biases. GPT offers real advantages over MBR. Most of them are minor, but if you must boot in EFI mode, the fact that GPT is the standard for EFI-mode booting is not a minor advantage; it's a major one.

Windows (uniquely) requires GPT to use EFI

My understanding is that this is a requirement of the Windows installer, but it's possible to get Windows to boot in EFI mode from an MBR disk by using various tricks, like converting from GPT to MBR after installing the OS, or by replacing a BIOS-mode boot loader with an EFI-mode boot loader without changing the partition table type. That said, my knowledge of precisely how best to achieve this goal is foggy, since I've only read about it; I've never done it myself. I've read about this only once or twice in several years, so this is not something that's done very often.

Can the EFI system partition be a logical partition ?

In theory, it shouldn't matter. In practice, I don't know. Booting in EFI mode from an MBR disk is such a corner case that there's a small but non-negligible chance that literally nobody in the world has tried using an MBR logical partition as an ESP. Certainly I haven't tried it. Please keep this in mind if you decide to pursue booting in EFI mode from an MBR disk -- your odds of getting expert (or even advanced amateur) help drop to near zero should you run into problems.

Overall, my recommendation is to look into MBR-to-GPT conversion tools. Chances are you'll be able to convert from MBR to GPT without trouble. The result will be less likely to cause problems than trying to boot in EFI mode from an MBR disk. You should, however, back up before doing this. That would be my advice even without an MBR-to-GPT conversion in the picture, though; mucking with boot loaders can be as risky as mucking with partition tables, so trying that without a backup is inadvisable, too.

Ordinarily, I would be happy using UEFI, but at the moment I'm playing around with Linux live USBs, which will only show up in CSM mode (they do work in CSM mode).

I can't be 100% positive of this, but I suspect this is a case of the Linux USB drives being improperly prepared. Most modern Linux distributions support booting in EFI mode, including booting their installation and emergency tools in this way; however, some tools, like YUMI, omit the EFI boot loaders when they convert .iso files to USB boot media. The solution to this problem is to use another tool to prepare the boot media, or possibly to adjust options in the tool you are using. See this page of mine for more on this subject.

If you're using an old or obscure Linux live medium that lacks EFI-mode boot support, you may be able to add it yourself by installing a suitable boot loader and configuring it manually. This is likely to be a hassle and reqires a deeper-than-average understanding of Linux boot loaders. My Web page on this subject may be helpful, but you'll need to read at least the first couple of sections and the section(s) on the boot loader(s) you intend to use. You'll also need to be able to parse the medium's existing boot loader configuration so you can "translate" its configuration to your new boot loader.

Another option is to switch boot modes whenever you want to boot the Linux media. How you do this varies from one machine to another. In most cases, you can permanently set an option in the firmware so that it supports both EFI/UEFI and BIOS/CSM/legacy booting. With that done, the computer's built-in boot manager should show options to boot Windows or from the USB drive. (The built-in boot manager is usually accessed by pressing Esc, Enter, or a Function key at boot time; but details vary from one computer to another.) Alternatively, you could install my rEFInd boot manager and edit its refind.conf file so that scanfor is uncommented and biosexternal is among its options. This should cause rEFInd to show options for booting USB drives in BIOS/CSM/legacy mode, if they're inserted when you boot. (In some cases, you may need to press the Esc key in the rEFInd menu to get these media to show up, or uncomment the uefi_deep_legacy_scan option in refind.conf.)

One final comment: If Secure Boot is enabled, this feature can limit your options. Major Linux distributions like Ubuntu, Fedora, and OpenSUSE all support Secure Boot; but more obscure distributions might not. Likewise, if you install your own boot loader or use rEFInd, you'll have to either disable Secure Boot or jump through extra hoops to get it working, as described on this page of mine. You can usually disable Secure Boot without enabling the CSM, but I've heard reports of EFIs that link the two options.