Okay, so, let's sort out your scenario.

The first thing to understand is that the witch, medium, and seer all start as baseline Gifted. They can "sense" flows of essence. This pretty much means any abnormal essence can be sensed, like the presence of a spirit, the use of essence based abilities, and maybe the release of essence due to intense emotion. At best, they should get a vague sense of direction and intensity. Kind of like how your sense of smell is when something is mostly diffused but still detectable. Imagine playing a game of "find the smell." The unchanged True Immortal is a mundane for all intents and purposes right now, so the only way he would sense spiritual activity is if he witnessed a physical manifestation/phenomenon.

Now, as far as I know, magicians has no invocations that let them "see" spirits. Communion can do something but it's more like an intense awareness and connection to everything, imagine if everything in the space was you and you were aware of everything in you. I gave Insight an effect that lets the magician "see" spirits. But, in your scenario, I would say that the witch would need a difficult Task to sense the presence of a docile spirit, an easier Task to sense it when it got agitated, and then no Task when she's in such close proximity to a spirit about to take action in the real world. You'll note that it takes Essence to manipulate the real world. However, even if she did have Spirit Mastery, she would need the subtype to be Spirits of the Dead in order to directly summon or "master" the spirit.

The seer is in a fun camp. Without Mindsight, she has no capacity to sense the spirit's presence beyond her natural Gifted sensitivity (included in WitchCraft but not Armageddon). Like the magician, she gets the difficult Task to sense the presence when docile, easier when it's agitated, and dead obvious when it's taking action that uses Essence. However, this is an interesting juncture. It seems to me that Mindsight would best be expressed as a constantly active power at a low level. You could do something like that. If not, then the seer would need to ping that a spirit or something is in the area, activate Mindsight, then start looking around with her newly upgraded sensitivity.

The medium is similar to the seer: she must have and activate Death Speech 1 or just have Death Speech 2 (this is always on). This, of course, means that an amateur medium would have to ping that there is even a spirit (in the same way as the seer or witch) then activate, then act accordingly. This puts her very close to the seer in functional ability with the notable exception that she very likely has this power (the seer has a wider and more varied selection of powers, imagine one with Mindhands and Mindtime). However, even a moderately experienced medium (Death Speech 2) will be able to "see" spirits as if they were a normal part of life (note that this is not quite the same as having the senses of a spirit, but pretty close and very practical). The rest of your scenario requires some decisions on your part on the interactions of spirits and others.

The basic one seems to be very straightforward: if you can "see" the spirit, you can also "hear" the spirit. Maybe you're not so kind and you say that you can "see" the spirit but still cannot actually interact with it. Maybe you're like me and "see" and "hear" are inaccurate descriptions but the end results are the second option. Maybe you're still like that but the end option is the first. The details add verisimilitude but the end result is pretty simple: see = interact or see =/= interact.

So, when the spirit starts acting up, she starts expending essence to affect the real world. The gifted can sense that pretty clearly (WitchCraft, p 196, last paragraph, left column [note, I apply this universally, not just to magicians even though that's the focus in that section]), regardless of their training. Again, they can't exactly pinpoint the details, but they have enough of an understanding of the intent to be able to defend themselves and maybe even target the area most likely to contain the spirit. The seer could pinpoint the spirit by activating Mindsight. The medium could pinpoint her with Death Speech (1 requires activation, 2 is automatic). Meanwhile, the poor unchanged True Immortal is completely blind to the actions and tells of the spirit (remember, he can't sense essence unless it literally has a physical manifestation).

Let's start by defining the characters, because that is a hugely important step. Oh, look, WitchCraft courteously does that for us. Let's use the Wicce private eye (p106), Ghost Hunter (p116), and Seer Fugitive (p118); we'll use "A Typical Ghost" for the chinese lady (p. 268). Let's jump over to Armageddon and use the Soldier (p102) as a suitable stand in for the unchanged True Immortal. Check it out, your crew is defined for the purposes of the scenario. Remember, the skills and abilities are important defining factors since Unisystem is based on choice rather than class.

It's 8a on a Wednesday and our intrepid band of occult insiders is sitting at a diner having breakfast. The diner tends not to have much spirit activity, so when the Chinese ghost comes by, everyone with the Gift gets a chance to detect her (maybe a Difficult Perception Test at -2 or a Perception and Notice at -6). This means the witch has a 30% chance to notice a spiritual presence and the seer has a 20% chance (not much, we'll say they don't notice). The medium has Death Speech 2 so she sees the Chinese ghost come in and settle nearby. Meanwhile, you might want to give the unchanged Immortal a chance to notice the tells (like in poker) of his peers (Perception and Notice, situational awareness applies, at -2; he has a 10% chance of catching on). Let's go ahead and say no one with less than 50% chance notices so only the Medium is aware of the ghost. However, when breakfast is served, it contains unhealthy Western foods and this happens to be a peeve of the Chinese ghost. Nothing about the situation has changed yet and no one is sharp enough to pick up on the medium's tells regarding her presence. As the meal progresses, the ghost gets increasingly agitated that she is being ignored by everyone at the table and the medium is trying to be polite without disturbing the others. Eventually, she gets so angry that she is practically screaming at them. This changes the circumstance because the ghost is no longer in a docile state, it is now hostile and unconsciously releasing energy (not enough to count the points but you get the picture).

With the spirit now extremely agitated, the seer and witch should have another chance to sense her presence. Now, though, it's pretty obvious that she's not happy so lift the penalty. That means the witch is at 50% and the seer is at 40%; we'll say the witch notices but the seer doesn't. The dynamic quickly changes at this point because the witch can sense the spiritual energy and get a sense of the flavor, in this case anger. She is aware that there is an angry spirit in the immediate vicinity and if she has been exposed to this ghost, she might even recognize it. However, once she addresses the medium about this, everyone should be aware.

At this point, the Witch could decide to use Insight to see the spirit (if you let that fly). With the announcement of its presence, the Seer will activate Mindsight. All of this essence use frightens the Chinese ghost and she reacts angrily. When she uses her power of Animate to throw things about, the active use of Essence at this range requires Simple Perception Tests to notice (I would use Perception and Notice). Throughout all of this, the unchanged Immortal must rely on reading his friends' reactions and if he knows them, he'll recognize the magic words uttered by the Witch and the moment of concentration used by the seer followed by a wide-eyed hyperawareness. However, he can only tell the spirit's actions by the physical results and so he would notice the breakfast-ware flying about but would not have any idea of the source (though he could surmise telekinesis or a spirit; or if he wants to experiment, he could just take the top off the salt shaker and throw the contents in a wide arc to make whatever spirit is nearby recoil; our character archetype doesn't have occult knowledge so I don't think he'd know about that option quite yet).

Clarity of purpose. Clarity of conscience.