Sanity check on hot water line for dishwasher installation where there wasn't one before
The types of fittings that you linked are compression type fittings that are meant to be used where a compression ring is placed around a smooth copper or nylon copper water delivery pipe. Your setup is using the flex braided water connection hoses and really should use the proper type fittings.

If I was addressing this connection problem I would focus on replacing the coupling nipple at this connection:



(Picture source from question at this posting)

Note that your sink unit flex pipe ends with a 1/2" fitting.
Replace the nipple with a 1/2" NPT Tee fitting like this:



(Picture source from amazon.com)

Then adapt the branch of the Tee to fitting with a 1/2" female NPT to 3/8" MPT fitting to connect to the dishwasher supply line using a fitting that looks like this:



(Picture source from amazon.com)

Technically the only fitting threads that need the Teflon plumbers tape are those where the water seal is made with the threads themselves such as between the Tee branch and the 1/2" to 3/8" adapter as suggested above. The braided hose connections typically have a black or red rubber washer inside the female coupling nut that is designed to seal against the end of the mating male fitting and so no plumbers tape is really needed there. Some folks however seem to like to use the tape on all the threads such as where the nipple was installed in your pictures between the two flexible lines.