In your lecture example (chapter 14 Real Options example 1), the value calculated with the call option formula is $3.92m. This is considered project value “with” option. Value of option is $3.92 – $2 = $1.92.
However, in other information online, the value calculated with either call or put option formula states that the value should be value “of” option which is added to the NPV to get project value “with” option.
I get it now. When it is an option to expand, the additional expansion option benefit is added to the initial NPV. When it is an option to delay, the entire benefit of the delay is the new NPV.