Probability is used in weather forecasting. I will only highlight some examples due to my lack of knowledge in some areas.
Before initializing a forecast model the data needs to be assimilated. That means we need to somehow put measurements from ground based stations, satellites etc. into the model and fit this data to the grid. We then have an "atmospheric state estimate". This usually involves the minimization of cost functions. Typically the uncertainty of this "atmospheric state estimate" is used in the process of ensemble forecasting. An introduction to data assimilation can be found here at ECMWF. Every operational weather forecasting model has to do this data assimilation process. Since you specifically asked for names, here are a few: Global Forecasting System (GFS, probably the model you weather app uses), ECMWF model (European Center for Medium Range Weather Forecasts), Icosaheadral Nonhydrostatic Model (ICON, operational model of german weather service),High Resolution Local Area Modelling for numerical weather prediction (HIRLAM, operational model of some Scandinavian countries).
In an ensemble forecast the equations are solved as you stated in the question. However, there are certain aspects of the model that are varied to account for uncertainties. These aspects can include the initial conditions (including time, sometimes time-lag is used), variations of parameterizations (and there are a lot of them, basically for every process that can not be resolved by the model grid) and sometimes even multi-model ensembles. The model is then initiated many times to obtain multiple and different forecasts. Again, there is a nice introduction at ECMWF. Some ideas on how to visualize ensemble forecasts can be found here.
All of the above focussed on probability in the course of initializing a model. However, even if no ensemble is used and if we'd know the state of the atmosphere without error - we'd still have to rely on probability due to sub grid scale processes. A prominent example is cloud cover. Suppose we have a mean humidity $q$ in a grid cell (which is what the model gives us). Clouds will only form if $q > q_s$, where $q_s$ is the saturation humidity. Depending on other parameters like temperature a probability density function $Q(T,...)$ is assumed. A possible parameterization for cloud cover could be that the fraction of $Q(T,...)$ larger than $q_s$ defines cloud cover. I will not give examples for typical distributions used, since this is an active area of research. Once more I'd like to refer to ECMWF if you wish to see some details and examples of pdf's used.
Finally, I'd like to add that a main source of uncertainty in models (there is not necessarily probability used) is the parameterization of of convection (shallow and deep). Precipitation is highly related to humidity and convection and therefore large errors are often the result of parameterization.
I hope this helps somewhat.
Edit: Regarding the precipitation problem. What you see at google or the typical weather app is based on model simulations but most certainly not a model output. If google tells you "30% rain probability at your place" then this will most likely refer to a 30% chance that precipitation in an (by google unspecified) area exceeds a certain threshold. The certainty is
is generated by looking at the ensemble. The threshold is exceeded in 30% of the simulations. This is also the reason why a high resolution model is less accurate. It is easier to predict that it's raining 0.1 l somewhere in New York tomorrow than to predict that it rains 0.1 l in the central park tomorrow.
