Yes, they are
Each tradition has certain spirits that are better than others at certain tasks, while the specific spirits of each tradition are more suited to that tradition's views of their roles. So, while an air spirit of the shamanistic tradition may appear as a thunderbird, the air spirits of qabbalistic tradition may appear as a djinn. While their powers are similar, their roles are different.
The Core Rulebook (p.279) talks about this briefly but doesn't really explain how this restricts magicians, other than saying that they can only summon those five type of spirits.
Different aspects of the world (the elements) are important to traditions in different ways—the description of each tradition describes how each element lines up with the different types of spells, reagents, and spirits in their thinking, indicating which types of spirits and spirit powers they might be likely to call upon in particular situations. Each tradition’s description also includes the Attributes used in the Drain Resistance Test for spellcasters of this tradition.
The text from Street Grimoire (p.41) expands on this a little more:
Mages of that particular tradition may only summon the spirits listed with that tradition, and they are restricted in the tasks they can assign them. Assigning tasks outside the general area of their tradition will not receive a response from the spirit (for example, a Buddhist mage telling an air spirit to heal him will get no response, as air is a Combat spirit in that tradition, while the Health spirit is earth).
Do note, however, that this is talking mostly about bound tasks, such as Aid Alchemy, Sorcery, and Study or Spell Sustaining.
Despite the example used (asking for healing from a combat spirit), the only means that the six standard spirits can heal someone is if they have the Innate Spell critter power and take a Healing type spell, as no other powers available to them allow them to heal others. In fact, the power is only available to Spirits of Man. So, the only reason I think of about that example being used, is that the conjurer was asking for the air (combat) spirit to aid the conjurer on casting a Healing spell, which was already covered in the core rules:
Aid Alchemy, Sorcery, and Study: As a service, the
spirit can add its Force as a dice pool bonus to your Alchemy,
Spellcasting, Ritual Spellcasting (for spell rituals), and
Learning Tests if its type matches the spell’s category, as
listed under your tradition (p. 279).
Using this service means that when you want to cast a Fireball (a combat spell) and want to add your spirit's Force to your dice pool, a hermetic mage should ask help from a fire spirit, while a shaman should ask help from a beast spirit. Other spirits will not help on that task.
Historically, since the first edition of the game, any spirit can be used in combat, and this isn't something the developers would change without explicitly mentioning in the rules. There are many other tasks that they can perform other than fighting, though, and that's where each tradition's spirit types come in. In fact, they have been lifting tradition's restrictions on every edition of the game (example, hermetics could only bind elementals, while shamans could only summon spirits in 2nd edition), and it's one complaint from veterans, as that removes some of the flavor that all types of traditions had in previous editions.
Also, keep in mind that spirit task is not actually a game term, the term used is Services (tasks being used for technomancer's sprites), so there is no reason to assume that Street Grimoire author is trying to restrict all services available to your character with that clarification, especially given the wording used.
With that said, don't worry about it, all spirits can be used in combat, as that is a basic service that all bound and unbound spirits can perform, and none of the later-released books changed how this works.
Combat: You can have a spirit fight on your side in
combat. The entire fight counts as a single service.