There may be so-called "light-weight" distros, which come with a light-weight default package selection out of the box when installed. However, you can remove lots of unnecessary packages in a normal distros and replace your desktop environment with a less resource-hungry one. So you may be able to make any distro "light-weight" by aggressive package management. For instance, Ubuntu Server is basically the same thing as regular Ubuntu, except by default, there are no desktop environments installed, amongst other things. You can install all of the packages of regular Ubuntu if you want them. Or turn Ubuntu into Ubuntu server by uninstalling the desktop environment and other packages. In other words, the only significant difference is what is installed by default.
It is also worth noting that often distros are somewhat characterised by their package managers. The Arch Linux package manager, pacman, appears significantly faster when installing packages than apt from Ubuntu. If your PC has limited I/O performance, I would recommend Arch. For instance, I used Arch on a Raspberry Pi which boots off a micro SD card (SD cards are not really designed for random I/O workloads), and the difference is night and day when installing packages, when compared with Ubuntu.
To explicitly answer the question:
I will also use it to browse the web. So I have been adviced to "install a lightweight linux distro and it will fly on your new computer, because the OS is designed for old machines".
But, is that reasoning correct?
There may be some distros which come out of the box configured to be light-weight, so the reasoning may be correct. However, as I said, you can make a normal distro lighter with package management.
Even the oldest Intel Core i5 is not incredibly old, and 8GB RAM is not incredibly small. You did not mention graphics hardware. But I do not think your computer is that old that where will be that much difference between distros. Your biggest limitation is the mechanical hard disk, so a fast package manager may be desirable. Although a warning, as fantastic as Arch Linux is, I would not recommend it if you have little Linux experience. Occasionally things break because of the continuous release model, and you need to know how to fix problems or know how to identify the problem package and downgrade them.