Elbmek

The reason you're seeing Move there is because you're looking at threads (collections of posts), and not at posts within a thread.
The behavior you were expecting is exactly how it works for moving threads. Imagine a thread about military uniforms in a forum that's supposed to be about military vehicles, instead of the military dress forum. You could move the thread into the correct forum (and optionally, leave a link in the forum the thread was moved from that will be erased in 30 days to help folks find it).
The hierarchy, from smallest to largest, is as follows:
Posts (what I'm typing in now) Threads (collections of posts, in our case here called "Moving a Post" that contains all of our posts) Forums (collections of threads, in our case here, this thread is currently in the "Feedback and Ideas" forum, which is a child of Sales, Feedback, and Information category-type forum) You can move forums from within the ACP. This works by dragging and dropping forums in and out of parent and children containers.
You can move threads from one forum to another, anywhere within the hierarchy of forums you've created from the ACP. Out of the box, this is referred to "Move" and is what you had in your screenshot above.
You can move posts from one thread to another (or create a brand new thread consisting of the posts you're moving). Out of the box, this is referred to as "Split" and is what we were looking at earlier where you wanted to move a single post or subset of posts from a single thread into another.
Hope that helps!

I think the confusion stems from the fact that the "split" option for moderators has two inherent functions. In the current incantation of IPS, split is the single option you'd select for the following two scenarios:
Move selected post(s) in a thread to a new thread in the same or different forum; and Move selected post(s) into a different thread that already exists, regardless of what forum it's located in I think what @Elbmek is trying to do is the first option. One of his members posts a funny picture in reply to a conversation about something else, and he'd like to remove that post from that discussion and instead put it in a new thread in the funny post section of his web site.
From @Elbmek's perspective (and feel free to correct me if I'm wrong), keeping things simple would involve to separate options in the moderator toolbar. One that moved posts into new threads in a forum of his choosing, and a second button that stuck posts in the middle of a preexisting thread.
"Move" may be a better name for "split" in the moderator toolbar. While move exists for threads within a forum, it's got a different name at the post level within a thread. I think the names we have now are just a byproduct of the things that existed before IPS and how they worked in the past, rather than a reflection of any common sense.
I've been confused by the ability to merge two posts together into a single post, even when the posts are written by different people. That'd be a nice one to turn off entirely. Good luck fixing that mistake. 🙂
To give you a simple way to do what you're trying to accomplish, @Elbmek, tick the single post you'd like to move to your funny forum, click "Split" on the toolbar, make sure "new topic" is selected, and then choose your funny forum from the drop down list. The other options in that window are likely not things you're looking for, but you can do things like set the title of the new topic that will be opened in your funny forum, or lock the resulting thread so no one can reply, etc.
You shouldn't have to preselect a URL for anything unless you're putting all the funny picture posts into a single thread, in which case you'd copy the web address of the funny post picture thread and supply it for the existing thread option.

Also, @Elbmek, you could rename "Split" to "Move" on your forum for your moderators if you'd like to do so.
To do that, log into the ACP, go to Customization > Localization > Languages and click on the globe icon next to your installed language to "translate." You can then search for the word "split" and provided a translated word or phrase to make things easier to understand for your moderators. For example, you may want to change the System string "Split" to "Move Posts."
Thank you for your service, and for providing your feedback. It takes a bit of a distinguished and trained eye to point these things out sometimes, and while others here may not bother with such things and accept them as they are, I think you raise a wonderful point about decisions in the user interface that could use a bit of KISS.
Split doesn't do this. All subsequent posts will go wherever the poster replying decides to post them (unless you lock a topic to further replies). Split, in all honesty, just moves posts to either a new thread or a preexisting one. It does nothing more.
I think IPS uses the word split here because that's what other forum software was using when it was first written, and they kept the conventions at the time. One of the most common use case scenarios would be where in the course of discussion on a thread, the topics branch off. As an example, perhaps @Elbmek has members in a single thread talking about military uniforms and then someone adds a funny picture, and a series of replies are made about the picture that have nothing to do with uniforms any more.
In this case, you might want to "split" that conversation into two--keeping the military uniform discussion in one place, and the funny picture and anyone that has commented on the picture into a separate thread. In this context, "split" makes sense--the moderator is "splitting" off a series of posts. Yet, there are other reasons to move posts that will vary from community to community.