How to install WordPress in subdirectory?

You can run as many WordPress installations as you wish on a single domain name. Personally, I install each in a different folder, then select which one I want to run on the domain itself using this technique:
https://codex.www.ads-software.com/Moving_WordPress#Giving_WordPress_its_Own_Directory_While_Leaving_the_WordPress_Index_File_in_the_Root_Directory

All the rest remain active and usable, and are accessed by specifying example.com/folder as the URL.

What you propose will work, but you will, over the long term, find the approach detailed in that documentation easier to deal with, as it is just too easy to delete the wrong files in the root of a domain.

Another alternative is multisite which I have just started to use. Details are here:
https://codex.www.ads-software.com/Create_A_Network
https://codex.www.ads-software.com/Migrating_Multiple_Blogs_into_WordPress_3.0_Multisite

I think this answers my enquiry, but need to make double sure before I change anything on the server.

I am supporting the MySQL website of my client and updating it regularly with Dreamweaver. I want to suggest moving to WordPress and want to show them this by inviting them to have a look at a subsection of their own website. But at the same time I can’t risk disrupting what they already have.

So can I set up a subdirectory on their website and put little example blogs into that, whilst the top directory runs as usual? I.e. https://www.example.com runs as before, but https://www.example.com/theme1 and https://www.example.com/theme2 runs WordPress in separate subdirectories?

My preferred approach to addressing your needs would be to install WordPress twice, once in each folder, /theme2/ and /theme1/, rather than any of the methods discussed above.

An added bonus is that, if you decide that say /theme2/ is what you want to be your final production system, it is easy to make it look like the root (URL of just the domain name) and do a near instant switch from your current DreamWeaver site.