With respect, I believe Begin["`Private`"] is acceptable, and perhaps preferable. Begin["Private`"] establishes a context "Private`". This is fine, but if two packages do that they will share that context. Begin["`Private`"] establishes context HNotation2`Private`. This will be private to this package only.
The reason HTransR appears to be undefined is that it is defined within context HNotation2`Private`, which is not on the $ContextPath -- it can be accessed by its full name as HNotation2`Private`HTranR.
If you wish it to be accessible by its short nme, you could place a usage statement for it before Begin["`Private`"], or just define it there. Then it will be in the package context, which is placed on the context path when the package is loaded.
Begin["Private`"];. Afterwards, you will findHTransRin the context"HNotation2`Private`". Unless you let appear the symbolHTransRalready beforeBegin["Private`"];, you have to refer to it byHNotation2`Private`HTransRfrom outside that package. – Henrik Schumacher Feb 04 '19 at 17:58Begin["`Private`"]– Jason B. Feb 04 '19 at 18:05Begin["HNotation2 `Private`"];), but of course that is a bit redundant and error prone. Once again, sorry guys for the confusion caused by me. – Henrik Schumacher Feb 04 '19 at 18:29