Terminology of time series
I think these are often used colloquially as synonyms, but let's try to find the differences.
Each of them begins with &quot;Time Series&quot; (TS). So the difference lies in the three following terms. here with my interpretation:

Analysis - wanting to describe and understand characteristics the observed data coming from the generating function$^1$.
Forecasting - wanting to somehow predict the future output of the generating function.
Modeling - wanting to describe and understand the generating function itself, via an approximation.

So as you see, I think time-series analysis and time-series modelling are almost the same thing. I just feel the latter most likely involves really using a real model, with parameters. That doesn't mean specifically parametric or a non-parametric approaches.
I personally wouldn't draw a line between them in terms of &quot;uses statistics&quot; vs. &quot;uses machine learning&quot;, as your question suggests.

A natural progression in approaching a time-series problem (in my opinion) would be:

understand the data (TS-analysis)
understand the underlying function (TS-modelling)
test how well I have understood the function and its output (TS-forecasting)


$^1$
With the term &quot;generating function&quot;, I don't mean the strict mathematical definition, rather just the underlying process or system that is generating the observed data