Wikipedia gives you an overview of the classes in a classful network. Class A contains all addresses in which the most significant bit is zero, so all addresses from 0.0.0.0 up to 127.255.255.255. If you say "Class A IP address" you mean an IP address in the given range. If you say "Class A PUBLIC IP address" you mean an IP address in the given range which public.
Public IP addresses are globally routable unicast IP addresses. If you take a look at Wikpedia's IPv4 article or RFC5735, you can see that there are a few ranges in Class A that are not globally routable:
- 0.0.0.0/8 (0.0.0.0-0.255.255.255): only valid as source address in special cases
- Private network: According to RFC1918 the network 10.0.0.0/8 (10.0.0.0-10.255.255.255) is a private class A network.
- Loopback address: 127.0.0.0/8 (127.0.0.0-127.255.255.255) is for local loopback.
- Shared Address Space: 100.64.0.0/10 (100.64.0.0-100.127.255.255).
So if the private range is the one that should ideally be used in
private networks, while all the other addresses in the Class A IP
range will be Class A public IPs (which makes no sense to me),
But this is exactly what it is like. There are a few private network IP ranges. One of them is 10.0.0.0/8, which is a Class A network. Another one is 172.16.0.0./12, which is Class B. And private network range 192.168.0.0/16 is Class C. Any IP address (range) you are giving can be classified a Class A, B, C, D or E network. Classful network was first. So first all IP addresses where only put in these classes. But as things evolved there was a need for more granularity and new ranges. So CIDR was invented and someone took the unused range 10.0.0.0/7 in Class A and said that this range is not for public IP addresses anymore but for private network.
then why can't we simply say that the Class A IP range is from 10.x.x.x to
10.255.255.255 itself, because the others in the Class A IP range will be Class A public IPs.
Because this is wrong. You are mixing two stages of evolvement (Class A = classful Networking, public IPs = CIDR). According to Wikipedia IPv4 the first IP addresses had been divided up into 8 bits network and 24 bits host part. This was not flexible enough, only 256 networks maximum were too few, so someone invented classful networking with Class A, B, C, D and E. This second version of IP allocation replaced the first version. Again later someone invented classless inter-domain routing (CIDR). This third version of IP allocation replaced the second version. The classful network allocation scheme was not changed, noone changed Class A or any other class, the classful network scheme is just not used any more.
Class A range was defined to be the range 0.0.0.0-127.255.255.255. This was not changed and it will not change. There is no need to change it. Classful networking is not used nowadays, we are using CIDR now.
You might argue that the inventor of CIDR could have redefined Class A and introduced new classes. But this makes it unnecessarily complex when the same name "Class A" has different meanings depending on the IP allocation version used.