[H+] is shorthand for [H3O+]. H3O+ (hydronium ion) requires water plus a hydrogen ion from the acid. There aren't actually free H+ ions floating around. They are all attached to water molecules.
That's the definition of p[H]. The definition of the "real" pH is the -log of H+ activity. H+ activity is the concentration of H+ * a coefficient. In practice, the coefficient is considered to be 1, however there are cases (such as extreme concentrations) when that approximation isn't valid.