Without the \pause command, "On the first or third slide" has no specification for when to appear, which means it always appears. With the \pause command, it uncovers the list items one at a time until they are all uncovered, and then pauses again, which is why you have the repeat slide (since nothing changes between uncovering the last item and pausing).
The solution is to specify when you want the items after the list to appear. One method of accomplishing this is to surround it with an \uncover command:
\documentclass{beamer}
\begin{document}
\begin{frame}
\begin{itemize}[<+->]
\item On the first slide
\end{itemize}
\uncover<+->{
On the second slide
}
\end{frame}
\end{document}
Edit
In response to your edit, here's another way to solve your problem. You are using \pause in combination with the list option [<+->], which adds its own pauses and so results in double pauses sometimes. If you really prefer to not give up \pause, then you can fix your example's behaviour by using it exclusively:
\documentclass{beamer}
\begin{document}
\begin{frame}
On the first slide
\pause
\begin{itemize}
\item On the second slide.
\pause On the third slide
\pause\item On the fourth slide
\end{itemize}
\end{frame}
\end{document}
This also works in your first example:
\documentclass{beamer}
\begin{document}
\begin{frame}
\begin{itemize}
\item On the first slide
\pause\item On the second slide
\end{itemize}
\pause On the third slide
\end{frame}
\end{document}
The potential down side of this approach is that \pause stops everything after it on the current frame from appearing until the pause has passed, even if you specify that it should appear earlier. For example,
\documentclass{beamer}
\begin{document}
\begin{frame}
Here are important points:
\begin{itemize}
\item Point on Slide 1
\pause\item Point on Slide 2
\pause\item Point on Slide 3
\end{itemize}
\uncover<1-3>{
I want to emphasize this on every slide:\\ Always remember these points!\\ (But it only appears on the third slide)
}
\end{frame}
\end{document}
On the other hand, with less code you can achieve this effect simply:
\documentclass{beamer}
\begin{document}
\begin{frame}
Here are important points:
\begin{itemize}[<+->]
\item Point on Slide 1
\item Point on Slide 2
\item Point on Slide 3
\end{itemize}
Always remember these points!
\end{frame}
\end{document}
Section 4.3.3 of the beamer user guide (texdoc beamer on a Unix system) is called "Ways of Improving Compilation Speed". There are two suggestions there: to use the draft class option, and to use the \includeonlyframes{list,of,frames,to,process} command. This means that only frames whose label matches one in the list will get processes. The suggestion is to have \includeonlyframes{current} and to keep moving the current label from frame to frame as you work on different ones.
MWE:
\documentclass{beamer}
\includeonlyframes{current}
\begin{document}
\begin{frame}[label=current]
This frame will be included.
\end{frame}
\begin{frame}
This frame will NOT be included.
\end{frame}
\end{document}
Best Answer
No. There is no way to do that. At least, there is no way to do that with
beameror any TeX software I'm aware of. The pauses just are separate pages in the output PDF. When you display the slides, a pause corresponds to a change from one page of the PDF to the next. So, if you have n pauses on a slide, you need n+1 pages in the PDF. Remove the page breaks and you remove the pauses. These are not distinct things. That is just how the pauses are implemented in the PDF.EDIT
If you in fact want what Sigur described - that is, the first item on the first slide only, the second on the second slide only and so on - that is straightforward:
This puts the first item only on the first slide and the second item only on the second slide. However, it does this by using two pages in the output PDF: