What is the difference between repeatability, replicability and reproducibility?
The American Statistical Association (ASA) has developed Reproducible Research Recommendations. The purpose of these recommendations are to create transparent science that can be recreated independently of the data creators and analyzers.  These recommendations have been motivated by several high profiles examples of studies being either retracted or not repeatable. The ASA includes definitions for Reproducibility and Replicability:


  Reproducibility: A study is reproducible if you can take the original data and the computer code used to analyze the data and reproduce all of the numerical findings from the study. This may initially sound like a trivial task but experience has shown that it’s not always easy to achieve this seemingly minimal standard.
  
  Replicability: This is the act of repeating an entire study, independently of the original
  investigator without the use of original data (but generally using the same methods). 


I do not know repeatability fits into these definitions or if it has a formal definition. Also, this hierarchy seems to differ from yours. Reproducibility is using exiting data and recreating the same results using the described methods. Replicability is conducting a new experiment and reaching the same conclusions. 

On a personal note, I have tried to reproduce studies while working with a statistician and most life science journal  articles do not include enough details to exactly recreate analysis unless they include their script (e.g., Python, SAS, or R code used to analyze the data). We were trying to recreate simple linear regressions and ANOVAs to find case studies for undergraduate stats courses, but often got different regression coefficients and test statistics. For example, people often make assumptions about NA values or transformations that are not described in their formal writeup.

For examples of replicability problems, search for psychology reproducibility problems. Also, notice the inconsistent use of the r terms here.  

For examples of reproducibility problems, this lecture describes how a forensic statistician uncovered the Duke Cancer Scandal a few years back. Although outright fraud was occurring, there were other serious issues with the data analyzes.