That is not correct.
First note that the HTTP 1.1 datagram does not contain an IP address at all; that information is in the layer 3 datagram. The only field in the HTTP 1.1 header that is related to proxying is the authentication header, which indicates whether the proxy requires a login.
SOCKS occurs at layer 5 (below HTTP, but above IP), and involves creating a pair of circuits between source and destination, so it does not just pass on the datagrams it recieves; it fully reencapsulates them from layers 5 and below, so that the reciepient server doesn't even know at layer 5 that the client is using a SOCKS proxy at all. that header was removed in transit, and the layer 7 payload was placed in new transport segments, and in a new packet before being sent on to the remote destination.
It is possible to determine whether known proxies were used in a communication based on databases of IP addresses for known proxies, but there should be no explicit evidence in the datagrams passed between the proxy and the destination.
It is important to note that the end result of proxification varies by the adversary. Nation-states with infrastructure that can see large swathes of the web will likely be able to tell who you are, and their capabilities are well beyond the scope of these forums.