Clarification on Web Integration using Reverse Proxy

I am having trouble understanding where the installation of a reverse proxy would occur, specifically on which side of a web server.

The scenario is this. I want to be able to take a customer's subdomain, for example https://ift.tt/2H74BNs and "redirect" it to my web server to serve the content of the site. I do not want to use an actual redirect because to my understanding a redirect will show information about ports or IP address to the customer I would rather not share, so instead I am opting for a reverse proxy for this migration. I wish only to understand how the network would be structured and how an example request would flow through the process, not how to code it.

My question: Would I install the reverse proxy on the customers website (their web server)? If so, would the process of the end user request be similar to below:

End user sends request to view https://ift.tt/2H74BNs -> request gets sent to reverse proxy and only requires single configuration to proxy -> reverse proxy finds subdomain on a restricted environment -> restricted environment allows reverse proxy request due to restrictions and returns to reverse proxy -> reverse proxy sends completed request back to customer web server.

Thank you in advance

Attempting to learn reverse proxies to be used to connect to private or restricted web servers. I am self-taught, and tried using Apache and NGinx along with Stack Overflow for my initial research into this question.

No code necessary at this stage. I am trying to understand the process.