How to explain (non-/anti-) aromaticity in fulvene with the help of resonance structures?
TL;DR: You cannot assign aromaticity based on a couple of resonance structures. Penta-fulvene has negligible (anti)aromatic character, which is supported by computational and experimental investigations.



Introduction

Aromaticity is a complex and still not fully understood phenomenon. Active investigations are experimentally and computationally challenging. Unfortunately in schools and university it is often taught as something quite simple to fathom, which can be explained by looking at Lewis structures and counting electrons. This maybe true for many common compounds, but when you dig deeper you'll very soon find the limitations. (See the notes below.) It is certainly not helpful in the case of fulvenes.

Penta-fulvene's resonance and (anti)aromaticity

The resonance structures you have drawn are correct, but the set is missing one member, coincidentally the more important one. (Please see the notes below on resonance.) There are more, but those are with more charge separation and likely have only little contribution.  



In general you cannot judge one resonance structure on its own. In this case it is not helpful at all. In all resonance structures, the &pi;-system is fully conjugated, and delocalised over the whole molecule.
Penta-fulvene has C2v symmetry, and we see deviations in the single and double bond lengths. The values are from a quite extensive study on substituted fulvenes: K. Najafian, P. von Rague Schleyer, and T. T. Tidwell, Org. Biomol. Chem. 2003, 1, 3410-3417 (DOI: 10.1039/B304718K). Unfortunately, they use the unsubstituted fulvenes as comparison. From the abstract:


  The fulvenes (1a–4a) have modest aromatic or antiaromatic character, and are used as standards for comparison.


Another study basically comes to the same conclusion, see E. Kleinpeter, and A. Fettke, Tetrahedron Lett. 2008, 49 (17), 2776-2781 (DOI: 10.1016/j.tetlet.2008.02.137). Citing quite liberally from various parts and omitting any literature references:


  Fulvenes 1–4 have previously been synthesized (triafulvene 1, pentafulvene 2, heptafulvene 3 and nonafulvene 4), and were studied with respect to their dipole moments and NMR spectra. The 1H and 13C NMR spectra of triafulvene 1 (both protons and carbon atoms of the 3-membered ring moiety display resonances in the region of aromatic compounds) evidence a significant contribution of the resonance form 1b [aromatic charge separation]; the corresponding NMR spectra of 2–4, however, display typical olefinic compounds with strongly alternating bond lengths and only a small extent of charge separation (corroborated by the relatively small dipole moments).
  
  [...]
  
  Dependent on the criterion employed, 1–4 were reported as partially aromatic, non- or even antiaromatic.
  
  [...]
  
  [...] However, the expected partial aromaticity of the 3-membered ring moiety of 1 was not observed (vide supra).
  Similar conclusions can be drawn for the presence of partial aromaticity in 2: even if the occupation of &pi;C=C of the exocyclic double bond is lowest in the series (which can be realized with the participation of 2a, corroborated by the correct direction of the dipole moment), both ICSSs [iso-chemical-shielding surfaces] at ±0.1&nbsp;ppm [2: ICSS = −0.1&nbsp;ppm (5.0); ICSS = +0.1&nbsp;ppm (6.2)] are far away from reference benzene 7 [7: ICSS = −0.1&nbsp;ppm (7.2); ICSS = +0.1&nbsp;ppm (8.9)] or even from cyclopropenylium cation 6 [6: ICSS = −0.1&nbsp;ppm (5.9); ICSS = +0.1&nbsp;ppm (7.2)]—pointing to 2&pi; electron aromaticity. Again, if there is partial 6&pi; electron aromaticity in 2, due to the contribution of 2a, then it is only very small.
  
  [...]
  
  Compared with the corresponding fulvalenes, studied previously, which are genuine push–pull olefins and exhibit partial (anti)aromaticity in the corresponding 3-, 5- and 7-membered ring moieties (in the latter if structurally planar), the 3-, 5- and 7-membered ring moieties in fulvenes 1–4 reveal very small, if not negligible (anti)aromaticity only.


From all of the above I hope I was able to make clear how complex the concept of aromaticity is. Only because of thoughtful investigation and interplay between experiment and theory, penta-fulvene can be described as having negligible (anto)aromatic character.



Notes on aromaticity

The original definition of aromatic (gold book) only states is very broad and may include any and no compounds:


  
  In the traditional sense, 'having a chemistry typified by benzene'.
  A cyclically conjugated molecular entity with a stability (due to delocalization ) significantly greater than that of a hypothetical localized structure (e.g. Kekulé structure ) is said to possess aromatic character. If the structure is of higher energy (less stable) than such a hypothetical classical structure, the molecular entity is 'antiaromatic'. The most widely used method for determining aromaticity is the observation of diatropicity in the 1HNMR spectrum.
  See also: Hückel (4n + 2) rule, Möbius aromaticity
  The terms aromatic and antiaromatic have been extended to describe the stabilization or destabilization of transition states of pericyclic reactions The hypothetical reference structure is here less clearly defined, and use of the term is based on application of the Hückel (4n + 2) rule and on consideration of the topology of orbital overlap in the transition state. Reactions of molecules in the ground state involving antiaromatic transition states proceed, if at all, much less easily than those involving aromatic transition states.
  


Much more rigorous is Hückel's (4n + 2) rule and therefore includes much less compounds. The main problem here is, that its application is often taught carelessly or even wrong. When one considers whether a compound is aromatic or not, it is probably one of the worst rules to follow. For fulvenes it certainly leads to the wrong conclusions.
The main problem is that this rule often gets reduced to counting &pi;-electrons, but that is only a small part of it. Even if we include more recent developments and extensions of the rule, there is a lot more to it. (Originally only valid for a couple of hydrocarbons from which it was derived.) I like to encourage you to read up on the full definition (and links within) in the gold book:


  Monocyclic planar (or almost planar) systems of trigonally (or sometimes digonally) hybridized atoms that contain (4n + 2) &pi;-electrons (where n is a non-negative integer) will exhibit aromatic character. The rule is generally limited to n&nbsp;=&nbsp;0–5. This rule is derived from the Hückel MO calculation on planar monocyclic conjugated hydrocarbons (CH)m where m is an integer equal to or greater than 3 according to which (4n + 2) &pi;-electrons are contained in a closed-shell system. [...]


There is a more updated version on aromaticity in the gold book, which allows a more rigorous approach to the whole subject. Unfortunately it is not as simple as what was there before. You will need to understand a lot more about quantum chemistry, especially how to construct molecular orbitals. While Hückel MO calculations (which you could probably still do with a pencil and a [few] paper[s]) still provide a good entry point and approximation, it is more convenient using modern electronic structure programs and density functional theory (or similar) to elucidate aromaticity.
For the sake of completeness, here is the newer definition:


  The concept of spatial and electronic structure of cyclic molecular systems displaying the effects of cyclic electron delocalization which provide for their enhanced thermodynamic stability (relative to acyclic structural analogues) and tendency to retain the structural type in the course of chemical transformations. A quantitative assessment of the degree of aromaticity is given by the value of the resonance energy. It may also be evaluated by the energies of relevant isodesmic and homodesmotic reactions. Along with energetic criteria of aromaticity, important and complementary are also a structural criterion (the lesser the alternation of bond lengths in the rings, the greater is the aromaticity of the molecule) and a magnetic criterion (existence of the diamagnetic ring current induced in a conjugated cyclic molecule by an external magnetic field and manifested by an exaltation and anisotropy of magnetic susceptibility). Although originally introduced for characterization of peculiar properties of cyclic conjugated hydrocarbons and their ions, the concept of aromaticity has been extended to their homoderivatives (see homoaromaticity), conjugated heterocyclic compounds (heteroaromaticity), saturated cyclic compounds (σ-aromaticity) as well as to three-dimensional organic and organometallic compounds (three-dimensional aromaticity). A common feature of the electronic structure inherent in all aromatic molecules is the close nature of their valence electron shells, i.e., double electron occupation of all bonding MOs with all antibonding and delocalized nonbonding MOs unfilled. The notion of aromaticity is applied also to transition states.


Notes on resonance

I will not go into a lot of detail here, because bon did an excellent job explaining it in: What is resonance, and are resonance structures real? However, allow me to make one point very clear: You cannot treat resonance structures on their own. You always have to treat them as a set, a superposition. There is no such thing as a most stable resonance structure, as well as there is no such thing as one of these structures dictating the reactivity. From a pencil and paper approach you hardly ever can judge which structure is most important for the description of the total bonding. Also, from a simple Lewis-type drawing you can almost never judge the properties of the compound.