"ApisCP is an open-source hosting platform..."

The ApisCP website https://apiscp.com/ states, "ApisCP is an open-source hosting platform...", however the license associated with the project states,

Unauthorized copying of this file, via any medium, is strictly prohibited without consent. Any dissemination of material herein is prohibited.

As the freedoms to use, copy, modify, and redistribute are core to open source software and licenses, the language on the website stating ApisCP is "open-source" should be removed.

If you would like ApisCP to be available as open source software, please update the license with one approved by the Open Source Initiative.

You're conflating FOSS with OSS. This indeed is open-source, as you're viewing the source code in a public forum right now; however, it's not free open-source software. Licensing does allow those with an active license to freely install and use at their discretion, but not the unrestricted right to distribute the software as they see fit.

closed

According to Gitlab Open Source is, "software for which the original source code is made freely available and may be redistributed and modified." Gitlab also states, "Open source software is "libre" in that it is free to inspect, modify, and redistribute." That definition (ability to view, use, modify, and redistribute) is based on the Open Source Initiative's Open Source Definition.

ApisCP would be best labeled as, "Source-Available" as described by GitLab, "source-available code is proprietary (so not open-source) but you can view the source code."

Other organizations that include the ability to view, use, modify, and redistribute source code in their definition of Open Source (and reference the Open Source Initiative):

1. Linux Foundation: "The Open Source Definition is where you should start to understand how open source projects should be distributed, and what actually qualifies as open source."
2. Google: "Three important rights for an open source license to address are: the right to make copies (also known as reproduction), the right to modify and adapt (also known as the right to make derivative works), the right to distribute the original and modifications." and "The OSI collects a set of licenses which grant the relevant rights for open source projects. They also maintain a widely-accepted document which outlines what makes a license open source."
3. Red Hat: "Open source software is code that is designed to be publicly accessible—anyone can see, modify, and distribute the code as they see fit." ...and... "Open source software is released through a specific kind of license that makes its source code legally available to end-users...anyone can take the source code and distribute their own program from it."

Joining in on this discussion, I think the definition of open source you provide is certainly a valid one. At the same time, as long as there is no official, legal definition for what open source constitutes, nor is there any other definitive authority to set the term in stone, I don't see why one private organisation's definition of the term would be superior to another's. It is completely within the right of Gitlab or, indeed, the OSI to offer their definitions for the term, just as it is within the right of @msaladna and their company to propose and use theirs.

It is not too uncommon to see "open source" being used to describe a license which provides open access to source code, but forbids its distribution and/or unauthorised use, so I would argue it is just as valid of a definition as the one you support. After all, if openness is truly the goal instead of ideological or semantical purity, then making the source available under the given terms is a great step forward, especially when compared to other notable examples of similar software.

Let's overly simplify things...

Blesta vs WHMCS? If you had to pick one to call open source, which would it be? cPanel or ApisCP? Which one is open source?

We're discussing two interpretations of "open source": the semantic and the contrastive. This industry is colored with closed-source solutions - Plesk, CentOS Web Panel, DirectAdmin, and ServerPilot to name a few products that ship with obfuscated or compiled source. I like to refer to this as blackbox system administration, which is counter to the design of ApisCP. What's opposite of closed? Open, which is how we arrive at this contrastive usage.

You would prefer the semantic usage promulgated by your firm, whereas I am using "open source" as a contrasting adjective to the closed-source software prevalent in this industry. Now, several components of ApisCP do fall under FOSS licensed under MIT. They're available on GitHub.

Language changes, meanings morph to explain concerns. "Open-source" arose, as nuanced as it was, to contrast Netscape's usage of the word "free". "Open-source" in this situation arose, again nuanced, in contrast to its alternatives. "Source-available" may be an encyclopedic categorization of such software, but pedantry has no use here especially on an elevator pitch.

I've cross referenced TESS to see if OSI has a trademark on the term "open source" to which I saw no such indication. If this is in error feel free to let me know.