Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.