typec -- record

The difference is this: "the TypeC-TypeA cable" is intended to be pluged into regular TypeA USB host, which is always a host. The Type-A end has no means to communicate this to Type-C end, so the Type-C end is HARDWIRED to represent itself as "USB host". For this purpose the Type-C end has 56k pullup between VBUS and CC pin (some cables have 22k or 10k, which causes troubles sometimes). So when you plug the standard "TypeC-TypeA cable" into your MacPRO, the Mac thinks that the cable is the host, and host-host connection doesn't engage, and your HDD doesn't work. And since the HDD is not a host, the MacPRO can't work as device, so nothing happens.

The "TypeA female-TypeC male connector" is a different one. Its TypeC end has 5.1k PULL DOWN, which indicates to MacPRO that there is a device attached, and the MacPRO will enable VBUS, and your TypeA HDD will start communicating. The connection, however, will be less reliable because of reduced signal integrity - extra connector causes imperfection and degrade signals. More, many "TypeA female-TypeC male connectors" are made by people who have no foggiest clue what they are doing, and internal wiring is frequently very terrible in terms of super-speed transmission line quality. This is likely why you have an unreliable connection to HDD.

To make your illegal TypeA HDD to work with a single cable with a TypeC port, you either need to find another odd TypeA to TypeC male cable with 5.1k pull-down on one of CC pins, or to make your own using available breakout TypeC assemblies and very-very careful soldering. It might work.