I'm not sure what you mean by "below basically three methods existed to perform Gage R&R". There are a number of ways to estimate variance components, and there are multiple "issues" to be evaluated with respect to measurement systems. First a list of issues:
1. Discrimination (aka effective resolution).
2. Stability
3. Precision repeatability
4. precision reproducibility
5. Bias
6. Linearity
7. Accuracy
How you estimate each "issue" can be situation dependent.
For example, if you have a measurement device that lacks effective resolution, then it tends to round the data (too much) and therefore may give incorrect estimates. If the measurement system is unstable or inconsistent, then the conclusions you draw from the study may only be right for that inference space.
As I have already stated there are multiple ways to estimate the components meant to be estimated by the Gage R&R. For example:
R-bar/d2 using control chart method
Expected mean square estimates (EMS)
Restricted maximum likelihood (REML)
Bayesian
Each of these will likely give different values for each estimate.
You state "results obtained from them for the same data set will be significantly different". Please define significantly different. I don't see the results as significantly different. You would have the same rank ordering of components whether using JMP or the excel spreadsheet you shared.
I don't understand the question "Is there any way to repeat the results manually". Pardon my confusion, WHY? Why do you want to get the same results as your excel program?
You might enjoy reading Wheeler & Lyday "Evaluating the Measurement Process" or see here:
https://community.jmp.com/t5/Mastering-JMP-Videos-and-Files/Using-the-EMP-Method-for-Measurement-Sys...
"All models are wrong, some are useful" G.E.P. Box