A Noctua NH-L9i will not fit either one, at least not without extensive metal cutting. They seem to use Lenovos custom laptop coolers to get the job done.
That said, I would not purchase the 4770 as the CPU was released in 2013 and is part of rapidly aging DDR3 RAM era technology. It will just not work very well compared to a modern NUC.
The 6700T was released in 2015 and still has a year or two left. While the 6700T is the more interesting of the two with only a 35W TDP, modern grade NUCs draw far less at higher clocks and core counts.
For a web surfer or light office work machine, this is a decent option for now. However, I do believe the 6700T will not support Windows 11, which is more or less mandatory when Windows 10 goes EOL in October 2025.
For peace of mind I would rather look at something like this:
The $549 version comes with a 6 core 25W computer, runs circles around the other one, is slightly wider than the versions you have shown (39 mm vs 35 mm) and come with full company warranty. Choice is yours ultimately, but this PC I think could survive for something like 5-6 years in an office environment with no hassle, maybe a dirt cheap $30 DDR4 upgrade could be warranted though.
So, $550 now for five years or $200 now and another $200-$300 in two years?