It's best to fdisk and format /s the drive in the vintage machine. Then, temporarily move the drive to the modern machine, and copy \WIN98 directory from the Win98 installation CD-ROM, to the drive. Then, move it back to the vintage machine, boot, and run setup from the WIN98 directory.

Trying to partition and format in the modern machine can be an exercise in frustration if your old machine has limitations. The FAT32 type in the partition table changes depending on whether the BIOS has Int13h Extensions or not, for example.

The Win98 setup wizard also tailors itself to your hardware during install, based on probing for ISA devices, enumerating ISAPNP devices, consulting the BIOS/ESCD table, looking at PCI vendor/device IDs, and so on, so it really needs to run on the machine. Moving Win98 drives between machines causes a flurry of "New Hardware Found" prompts as the system rediscovers all the hardware, and can result in a randomly broken system.

Fully agree with the above.
Fdisk the hard drive in the system you want to use it in. then you know it's set up in a way the old PC understands.

As you seem to not enjoy running up the system, I'd suggest 2 partitions. That way you can permanently have the win98 install files on the drive should you ever need to reinstall. I'm just not sure if the USB reader will allow this

Another method is to use Rufus, change the image to MS-DOS and it will create the boot sector and copy DOS 6.22 files to the drive. Then just replace everything on the drive with your copy.

The requirements are:
Partition marked active
Boot sector (newer operating systems don't create the correct boot sector or partition ID)
And the system files: io.sys, msdos.sys, command.com

What FDISK does is install that boot sector and mark the partition active. Rufus can do that for you.

I have done this several times when replacing the old hard drives on our test systems (DOS, 95,98, NT) with compactflash at work.

If you are doing it this way because your floppy drive is not working or you do not have the ability to make floppies anyway, toss in a gotek drive.

Having said that:

If you have the possibility (meaning you have a way of doing it directly on the system - floppy, CDROM, etc..), that is the best approach.
If you are installing the OS on a completely different system (in my case, it is the same system) then it is better to just install from the target machine.

However, if there is special installed software you need from the original installation and you do not have the installation media and/or don't have the specific configuration details of that software, dealing with resolving the driver issues is a smaller problem.

Be mindful of your partition size limits. Use one of free partition manager tools to resize and align the drive after rufus is done.

I agree with the above responses that very best way is to do it all on the target system, but wanted to give you a direct answer to your question in case the above scenario is your reason for doing it the way you intend. That is why I have had to do it that way. This is NOT the best or even easiest way. It is just sometimes a necessary method.

If you do have at least the means to boot and use fidsk and sys the disk on the target machine, at least do that and then copy over your backed up drive. No rufus or partition manager involved.

Please do NOT use integrated partition tools in Windows Vista, 7, 8, 8.1, 10 and 11 to prepare partitions for a disk being used in a DOS based system, including Windows 95, 98 and Me.
First make megabyte-aligned partitions and do not care about the requirement for CHS-aligned partitions on the older systems.

Disruptor wrote on Today, 14:36:

Please do NOT use integrated partition tools in Windows Vista, 7, 8, 8.1, 10 and 11 to prepare partitions for a disk being used in a DOS based system, including Windows 95, 98 and Me.
First make megabyte-aligned partitions and do not care about the requirement for CHS-aligned partitions on the older systems.

Yes, 7 and up are better for making a SSD aligned partition for XP, mitigating one of the issues of it not being SSD aware.
For 4k sector 512e "advanced format" HDD, there will often be a jumper which will shift so that pre-7 partitioning will be aligned

Yep, partition and format using DOS tools in the actual DOS machine. If this is too much trouble for you, maybe you shouldn't be using DOS.

For DOS partitions, I typically set those up on the intended PC.

For setting up Windows partitions, I use AOMI Partition Assistant on my modern Win 10 PC.