Near the end of the chapter in which Momiji first appears, he put his arms around Tohru's waist, but didn't transform. And many of the events that transform the zodiac members look more like tackles or bumps than hugs. So what, specifically, are the specifications for a "hug" that will transform a member of the zodiac?
From my recollection, the determinant seems to be the non-zodiac member's arms going around the zodiac member's body. Intent doesn't seem to matter -- that's why you'll get tackles and bumps in there -- just the specificity of action. So that's how Momiji can hug Tohru -- it's just that she can't hug him.
Regarding this question, after reading to chapter 109 I think I've got a theory with some tangible proof. I will begin by compiling some odd incidents contained in the manga, which I think are pertinent, as well as general comments that help elucidate how the curse works. Note: with regard to my references, all volume numbers will to the English version, to the collector's edition which collects all chapters into twelve big volumes, because that's what I read. I will, however, include chapter numbers as well to avoid confusion.
Let me first give a brief overview of my theory. What I suspect, is that a qualifying "hug" has nothing to do with how a person's arms are positioned, but rather, has to do with body-to-body contact, and perhaps more specifically, torso-to-torso contact.
First, and importantly, there are offhand comments about crowds being dangerous for the Sohmas, which I think supports my theory that it isn't just proper hugs they need to worry about. In volume 7, chapter 76, Kyo says this:
In a crowd like that, I'd definitely get squashed and transform.
This kind of sentiment is repeated in a conversation between Hiro and Kisa in volume 8, chapter 88:
That aside, be careful while we're here, Kisa.
Huh? It's okay, Hiro-chan...! There are a lot of people here... But I'll be sure not to bump into anyone.
So it seems crowds are a major concern for the Sohmas. One wouldn't expect to get what is usually considered a hug under such conditions (with arms around the other person, etc.). On the other hand, bumping into people and "getting squashed" between people, as Kyo says, is something that could definitely happen.
Next, in volume 9, chapter 103 a conversation between Hiro and his mother, as they talk about Hiro's little sister, illustrates that just cradling a baby could trigger the curse:
Hold her, Hiro-chan. This is your chance.
What?! I-I can't do that! Hinata's a girl... So it's impossible for me!
It'll be fine! Mama will just hold the both of you!
The standard for cradling babies usually does involve holding the baby close to yourself and having part of their body rest against their own. I have a suspicion Hiro would probably be fine if he holds Hinata away from him and "Lion Kings" it, but that would definitely be bad form, possibly dangerous for the baby. My suspicion is based on the fact, as previously mentioned in the question, that Momiji put his arms around Torhu's waist when he first showed up, and also because Kyo has put both hands on Torhu's head many times (Yuki's done something similar as well, albeit not to Tohru, as far as I recall).
Now, I'll demonstrate some instances of "hugs" that are strange because they did cause a transformation or because they did.
In volume 8, chapter 87, there's this odd picture of Kyo giving Tohru a one-armed hug and seemingly putting his cheek against the top of her head. He doesn't transform in spite of this. It's unclear how much they're touching here, but I think the panel on the next page indicates Kyo was keeping his distance a bit. This is why I think him not transforming here doesn't disprove my theory.
Here's another odd scene where a transformation isn't triggered: Yuki pulls Tohru to his side and has one arm around her, but nothing happens. I think this helps further define what kind of hug is necessary. There's a lot of contact here, but side-to-side isn't going to cut it. That being said, the image could also be a little misleading, and it might be possible Yuki was careful not to come into contact with Tohru. It is pretty hard to tell in this case.
Now here's a scene where Momiji hugs Tohru and does transform. I only include it because the other answer suggests that the non-zodiac has to put their arms around the zodiac to trigger a transformation, and I think this clearly indicates that is not the case:
Now, here's what I think is the most informative scene, and it comes from volume 10, chapter 109:
This first picture shows Kyo really pushing the limits to the boundary of what he is allowed to do. He's pretty close to Tohru, he's got his arms around her, and his chin might even be resting on her head (hard to say for sure). But his body is not in contact with hers. Now here's the next part, where he does cross the boundary:
I was a bit baffled by this part at first, because he's clearly hugging Tohru, and yet doesn't seem to transform. However, it just so happens that this is where the author chose for the scene to end. A post-chapter picture does seem to indicate Kyo did transform:
So this is why I think this particular scene is so important to answering this question. It shows that arm placement isn't enough, and actually shows how close Kyo can get before he transforms. This scene alone gets very close to defining the exact limitations of a hug that will transform a member of the zodiac.
I think that all this information suggests that body-to-body contact is the key here, and not anything to do with how the arms are positioned. But Regardless of whether my theory is accurate, I think that by highlighting these cases I can help readers draw their own conclusion.