This Peh is referred to as the "Peh Shavur", or "Broken Peh". You can see the Peh Shavur in this old Yeriah I own:
Let's use reverse chronological order, that is, first understand the modern Peh. The source to the modern Peh is the Mishna Berura, who clearly rules that you should avoid using the broken Peh:
הג"ה ולא כמה שנהגו איזה סופרים לעשות עקב מבחוץ בצדה ... כי הוא ממש אות שבור. ובאמת צריך להיות עגול מבחוץ כמו שכתבנו ... ומה שנהגו כך מפני שאינם יודעים ההרגל לעשות לתפוס הקולמוס באלכסון ולהמשיכה מעט לאחוריה הקולמוס בפנים ... עכ"ל ספר כתיבה תמה בקיצור לענינינוThe Mishna Berura quotes the Sefer Ketiva Tama as the source of this ruling, claiming that the broken Peh is a mistake that should be avoided. When the Mishna Berura was first printed in the late 19th century, this ruling created a big controversy and debates in communities that had a long-standing tradition of using the broken Peh, and many Sofrim continued to write it in their old way for many years.
Then it came the First and Second World Wars, and many of the old scribal traditions were forgotten - including the old Peh. Following the wars, the rulings of the Mishna Berura became even more influential in communities around the world and the next generation of Sofrim relied heavily in the Mishna Berura's take on the Hebrew letters, effectively ignoring the controversies surrounding this ruling. That's how the modern Peh became the standard Peh in all subsequent holy scrolls.
But if you look at pre-war Torahs and Tefillins, you will often find the old Peh, specially in scrolls of eastern Europe and Russia.
For almost all readers, this is just a history lesson. But for the Chabad readers, this post is a eye-opener. The Alter Rebbe, author of the Tania, clearly writes that the broken Peh is a must and all Chabad sofrim have kept this tradition even after the wars. Thus, the Chabad communities have their own version of the Ktav Ashurit, which is different than the Ashkenazi, Chassidic and Sephardic scripts. This is known as the ktav Chabad, and the broken Peh is one of its signature characteristics. See the full Aleph Bet Chabad below:
You can also see the special Kuf, Mem Sofit and Tet prsent in this Ktav.
The Ktav Chabad is exactly the same Ktav used by the communities of Eastern Europe and Russia before the war. So the Ktav Chabad has survived the war to become one of the last standing old-European scripts in use, remarkably. Now you understand the tradition among the Chabad Chassidim that their Ktav is the most accurate and that when Moshiach comes it will become the standard script for all Jews. Perhaps it will.
UPDATE: In reponse to the questions raised in the comments thread, I did some further research and here's what I came up with. My source is Sofer Lipshitz, one of the most knowledgeble Sofrim I know, who happens to be Chabad. The real Old Peh, which is the modern-day Chabad Peh, ideally should have a smooth hunchback and not a real step - see this picture (note that the Chabad Aleph Bet picture above is not very precise):
However, there are many different versions of this hunchback Peh and some of them are a bit less precise then others. The Peh of my manuscript is one of these less precise Pehs - it's more than a slight hunchback and it really looks "Shavur", and perhaps this imprecise version of the Peh prompted the Ktiva Tama to protest against what he considered to be a "broken Peh/ Peh Shavur" and the Mishna Berura agreed with his claim. But as Zalman and Jskarf mentioned, it's very likely that the Mishna Berura didn't have any complaints against the precise version of the original Peh, which is roundish in the outside. Still, the "new and improved" Peh became extremely popular and it now our generation's standard Peh. In regards to the Chabad Sefer Torah of my Shul, it turns out to be that it is also an imprecise old Peh - not a slight hunchback but a very clear broken Peh. This is a small imprecision of the author of this Torah; the Chabad Peh should be roundsish in the outside. Yudi sent me the Peh of his shul's 120 years-old Sefer Torah and it seems to me that this is a perfect old Peh; not broken and round in the outside (click to enlarge):