To enter the second version above I used the Basic Math Template to get a "fraction" "Ctrl +/" then type the top and bottom of the "fraction: using the Basic Math Template. The instructions you provided will "correctly" enter the pde BUT it does not appear in the workbook the same as the example above.
When using the differential template I get:
TransportEquation2 = \!\(
\*SubscriptBox[\(\[PartialD]\), \({t}\)]\(u[x, t]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \({x}\)]\(u[x, t]\)\) == 0
Which, while it works, does not replicate the style of entry shown in the video that I am referencing.
However, I have found that if you right click the input line and select "Convert To" from the drop down and "TraditionalForm" from the sub menu you get an input that looks like what I reference above. I think that this is what is going on. The video presenter is "cleaning up" the input. I have seen this a few times and while it is great for making pretty presentations, it can really mess up those who are just trying to learn how to use the software. Following an example when how to construct the input is not clear is very discouraging.
Thanks