Following the question in subject, I guess there is a slight difference between NAT and the others.
Is it that NAT translates a complete private network, as SNAT\DNAT translate per package?
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"NAT" is a collective term for various translations - usually it's actually NAPT (involving the transport-layer port numbers as well).
Source NAT translates the source IP address, usually when connecting from a private IP address to a public one ("LAN to Internet").
Destination NAT translates the destination IP address, usually when connecting from a public IP to a private IP (aka port-forwarding, reverse NAT, expose host, "public server in LAN").
DNAT: Traffic from address IPs is forwarded to a specific internal IP. or redirects the incoming traffic for some IPs or Port to particular address of your choice.
SNAT: matches for all traffic from a specific network(internal Network) to the output interface (address source used as source for the packets that match, also the output address of your network).