Usually a model is considered to be high-dimensional when $n \ll p$, where $n$ is the number of the observations and $p$ the number of the variables/features (e.g. Bühlmann and van de Geer, 2011). However, you can find other definitions in the literature, as in Belloni et al. (2018) Ref. They write:
High-dimensional models are characterized by having a number of unknown parameters that is not vanishingly small relative to the sample size.
So when is a model high-dimensional? Is there a rule of thumb or some criterion, which defines that a model is high-dimensional different than $n \ll p$? I know that there is similar question on StackExchange (Ref), but this post is relatively old and the field of high-dimensional statistics/ machine learning developed in the last ten years.