You are correct that the term "Big Bang singularity" is considered outdated in modern cosmology. The reason for this is that the singularity refers to a point of infinite density and temperature, which is mathematically problematic. In the early stages of the universe's expansion, the laws of physics as we currently understand them break down due to extreme conditions.
Instead, cosmologists often refer to the initial stage of the universe's expansion as the "Hot Big Bang" or simply the "Big Bang." This terminology emphasizes the concept of an initial hot, dense, and rapidly expanding state of the universe, rather than a singular point. It signifies the beginning of our observable universe and the subsequent expansion and evolution of space, time, and matter.
The current understanding is that prior to the Hot Big Bang phase, the universe underwent a period of rapid inflation, which is a hypothetical phase of exponential expansion. However, the exact details of what triggered the Hot Big Bang or the mechanisms during the early stages of the universe's existence are still subjects of ongoing scientific research and investigation.