Consider the second tentative statement by the OP, slightly modified,
$$\forall \theta\in \Theta, \epsilon>0, \delta>0, S_n, \exists n_0(\theta, \epsilon, \delta): \forall n \geq n_0,\;\\P_n\big[|{\hat \theta(S_{n}}) - \theta^*|\geq \epsilon \big] < \delta \tag{1}$$
We are examining the bounded in $[0,1]$ sequence of real numbers
$$\big\{ P_n\big[|{\hat\theta(S_{n}}) - \theta^*|\geq \epsilon \big]\big\}$$
indexed by $n$. If this sequence has a limit as $n\rightarrow \infty$, call it simply $p$, we will have that
$$\forall \theta\in \Theta, \epsilon>0, \delta>0, S_n,\,\exists n_0(\theta, \epsilon, \delta): \forall n \geq n_0,\;\\\Big| P_n\big[|\hat{\theta(S_{n}}) - \theta^*|\geq \epsilon \big] -p\Big|< \delta \tag{2}$$
So if we assume (or require) $(1)$, we essentially assume (or require) that the limit as $n\rightarrow \infty$ exists and is equal to zero, $p=0$.
So $(1)$ reads "the limit of $P_n\big[|\hat{\theta(S_{n}}) - \theta^*|\geq \epsilon\big]$ as $n\rightarrow \infty$ is $0$". Which is exactly the current definition of consistency (and yes, it covers "all possible samples")
So it appears that the OP essentially proposed an alternative expression for the exact same property, and not a different property, of the estimator.
ADDENDUM (forgot the history part)
In his "Foundations of the Theory of Probability" (1933), Kolmogorov mentions in a footnote that (the concept of convergence in probability)
"...is due to Bernoulli;its completely general treatment was
introduced by E.E.Slutsky".
(in 1925). The work of Slutsky is in German -there may be even an issue of how the German word was translated in English (or the term used by Bernoulli). But don't try to read too much into a word.
I see various areas you should have a look into:
Here you should understand the most common continuous probability distributions (e.g. normal distribution, t-distribution) and the most common discrete distributions (e.g. binomial distribution and geometric distribution). You should also understand how they are related to each other, e.g. a t-distribution converges to a normal distribution if n goes to infinity. You should also understand concepts like conditional probability and Bayes' theorem and you should have a look into random processes, e.g. random walk.
- Basics of inferential statistics
You should understand the basics of inferential statistics and statistical testing. In statistical testing p-values and power of tests is important.
Linear algebra is one of the most important mathematical concepts for statistics. Important concepts are e.g. the inverse and the transpose of a matrix. You should also be able to calculate with matrices, e.g. multiplication.
- Regression and econometrics
There are three different areas of regression analysis: Cross-sectional regressions, panel data and time series analysis. You should go through all of the three areas. Time series analysis might be the most important area of this three areas for practitioners as it is used for forecasting.
- Machine learning algorithms
After having an overview of the different areas of machine learning you should have look in some of the most common supervised machine learning algorithms (e.g. regression and classification) and the most common unsupervised machine learning algorithms (e.g. clustering, cimensionality reduction and anomaly detection)
- Coding with statistical software
R and Python are the most widespread languages for statistical computing. If I were you I would choose R as you need less pre-knowledge in object-oriented computing for using it.
Best Answer
It's all about being able to show a potential employer that you have the skills they are looking for. A degree from a college is one piece of information that an employer can use for that, but not the only thing (nor does it necessarily translate into real world skills).
For me as a hiring manager even more important than that is experience and hands on examples. If you want to work in data analysis or machine learning my advice to you would be to do as much data analysis and machine learning work as you can. Start a blog, open a Github account, compete in competitions like on Kaggle. Depending on where you live, find a meetup, hackathon, etc.
Not only will you learn a lot from those experiences, you'll also meet a lot of people in the field and generate some examples of work that you can show an employer.