I learned that spontaneous reactions are reactions whose activation energies can already be overcome with the temperature they're already in. Since the temperature they're in already provides enough energy to overcome the activation energy, additional input of energy is not necessary and so it is a spontaneous reaction. So if this is true, why are some reactions like combustion reactions considered spontaneous if the temperature alone wasn't even enough to get over their activation energy? Is this definition of spontaneous wrong?
Clarification on the definition of spontaneity must be considered first, as it's meaning is often misinterpreted (and through colloquialisms is further confused). In order to do so, we must separate reaction concepts into thermodynamics (where free energy and spontaneity are discussed) and kinetics (where activation energy and rate are discussed).
Thermodynamics: looking only at the reactants' and products' Gibbs free energies, and the difference between them ( &Delta G ), a reaction is defined as (a) spontaneous if the difference is negative, (b) at equilibrium if zero, and (c) non-spontaneous (a sign that it is spontaneous in reverse) if positive. Thus, thermodynamics is interested only in which form is more stable energetically: the reactants or products. No details about the path (see kinetics below) is included. Check out Khan Academy's "Gibbs free energy and spontaneity" page for some great figures and further detail.
Kinetics: interested in the pathway between reactants and products, and how much energy change is required to obtain an arrangement that leads to one or the other. In other words, this is where the activation energy becomes important, as it is the energy barrier that must be overcome before a reactant could even think about becoming more stable. This barrier impacts the rate at which reactants and products can be formed. A "spontaneous" reaction in kinetics would be a "fast" reaction, one with a nearly zero or negative activation barrier (no representative "hump" in those internal reaction coordinate diagrams).
In summary, thermodynamics is interested in stability of reactants or products, and is not directly discussing the path by which it would arrive at one or the other. Hence, a reaction can be "spontaneous" and still not occur without further prompting because it is kinetically too difficult (read "high activation energy") to get there. However, a "spontaneous" kinetic reaction implies the path is "downhill" (so to speak) and thus is also a "spontaneous" thermodynamic reaction.
Following that understanding, you mention temperature, which is where I believe the confusion builds as it is mixed with the "spark" in combustion. Temperature does effect the Gibbs free energy by the following relationship:
ΔG = ΔH−TΔS
Thus, reactants can be "pushed" with temperature to become products by adding the necessary energy to a non-spontaneous or equilibrium system to breach the activation energy "hump," and be "spontaneous" at that temperature. In other words, they can indeed be triggered just by increasing the temperature. Examples of this are found in spontaneous combustion events as temperatures are slowly raised to a point at which the energy barrier is breached, and reactants are becoming products while releasing heat (because these spontaneous reactions are exothermic, recall negative &Delta G). Heat release then causes surrounding reactants to overcome the same barrier energetically, and result in what we know as fire/burning events.
What about a spark, then? A spark contains the energy needed to begin the combustion process (e.g. from gasoline and oxygen to carbon dioxide and water) but in a sudden shock that triggers this cascade of heat release much more rapidly. A CDC study illustrated the impact of this latent heat given off by the initial sparked reaction by observing a stoppering of combustion by removing the heat with conduction (i.e. blowing out a candle).
So, all in all, "spontaneity" as a definition perhaps isn't wrong, but is used in various circumstances to describe different processes; here, either stability between states or rate at which the states will change.
Be sure to check out the other related stack exchange page mentioned by Mithoron in the comments, with a great answer regarding spontaneity by Brian and Melanie Shebel. Covers more detail about the thermo/kinetic descriptions, whereas here we cover a more broad scope to address your questions about "spontaneity" in general.