welcome, if you are new, it is recommended to start with the 'official' STM core
https://github.com/stm32duino/wiki/wiki
this core supports (much) more boards and get something like a Nucleo board with more resources, check on the core repository
https://github.com/stm32duino/Arduino_Core_STM32
^ e.g. Nucleo F401RE or Nucleo F411RE in the list
or like
https://www.adafruit.com/product/4382
to start with.
if you insist on '3rd party' ones there are many too,
https://store.micropython.org/product/PYBv1.1
https://www.olimex.com/Products/Duino/S ... e-hardware
https://www.olimex.com/Products/ARM/ST/
https://www.aliexpress.com/wholesale?ca ... =stm32f401
but that these are less recommended for 'beginners' because you may be on your own if you stumble and is not familiar with stm32 mcu platform.
libmaple ('roger's core') these days is considered 'for experts', and if you insist do review the wiki
https://github.com/rogerclarkmelbourne/ ... STM32/wiki
and check other reading materials here
viewtopic.php?f=2&t=3
viewtopic.php?f=2&t=301
edit:
the technical reason for what you are seeing is because stm32f103c8 don't have a "DFU bootloader", check in the wiki.
as like discussed, Nucleo F401RE or Nucleo F411RE has more support and with the 'official' STM core
https://github.com/stm32duino/Arduino_Core_STM32
https://github.com/stm32duino/wiki/wiki
they are easier to flash, has more memory and flash and is less "distressing" for beginners.
Otherwise, get a board in the stm32f4xx series like those mentioned prior, those have the DFU bootloader built in *on chip*, all of them have it
if you get the Adafruit board like mentioned above, Adafruit made a nice tutorial about it:
https://learn.adafruit.com/adafruit-stm ... er-details