Triangle counting in hypergraph streams, including both hyper-vertex and hyper-edge triangles, is a fundamental problem in hypergraph analytics, with broad applications. However, existing methods face two key limitations: (i) an incomplete classification of hyper-vertex triangle structures, typically considering only inner or outer triangles; and (ii) inflexible sampling schemes that predefine the number of sampled hyperedges, which is impractical under strict memory constraints due to highly variable hyperedge sizes. To address these challenges, we first introduce a complete classification of hyper-vertex triangles, including inner, hybrid, and outer triangles. Based on this, we develop HTCount, a reservoir-based algorithm that dynamically adjusts the sample size based on the available memory M. To further improve memory utilization and reduce estimation error, we develop HTCount-P, a partition-based variant that adaptively partitions unused memory into independent sample subsets. We provide theoretical analysis of the unbiasedness and variance bounds of the proposed algorithms. Case studies demonstrate the expressiveness of our triangle structures in revealing meaningful interaction patterns. Extensive experiments on real-world hypergraphs show that both our algorithms achieve highly accurate triangle count estimates under strict memory constraints, with relative errors that are 1 to 2 orders of magnitude lower than those of existing methods and consistently high throughput.