Question about cables needed for using a Macbook Air 2013 OR Macbook Pro 2020 laptop with iMac 2013 AND Thunderbolt Display screens
Your question(s) are answered in this existing post:
What are the requirements to use iMac screen as a second monitor - Target Display Mode?

What I would like is to be able to hook up EITHER my Macbook Air (for personal use) OR my Macbook Pro (for work use) and use the laptops as the main 'hubs', linked to two large displays for both from the iMac and the Thunderbolt Display, so that I can transport my stuff easily to/from the office if I need to (where I can hook into another display at work).

Laptops are not &quot;hubs&quot; of any kind so, it's not clear what you're attempting to do here.  If you're attempting to sync all your work/personal things between the laptop and the desktop, then it's a different topic altogether (a different question) and easily solved with a Cloud service (OneDrive, iCloud Drive, etc.)
As for the cables - what you want to do is really not feasible.  First off, you are limited to Thunderbolt because that's what the 2013 iMac requires.  However, each notebook has a severe limitation to accomplishing what you want to do.  Let's look at them individually and not as a part of a larger puzzle:

2013 MacBook Air

Single Thunderbolt port / only supports one external display
To get a second display, you'll need a USB Video Adapter (it's like installing a second video card in a desktop machine)


2020 MacBook Pro

No longer supported for Target Display Mode to the iMac



So, right out of the gate, your MBA can't support the two displays and while your MacBook Pro supports up to 2 external displays it can't support your iMac as a second display (see the above link about Target Display Mode).
So, to connect your MacBook Air to either the iMac or the Thunderbolt Display, your need a Thunderbolt Cable.  That's assuming it's the older Thunderbolt Display.  If it's the new LG Ultra Fine, you'll need a Thunderbolt 3 to Thunderbolt 1/2 (Legacy) Adapter.  However, it's probably &quot;too much&quot; monitor for that legacy MacBook Air - it'll work poorly.
As for your MacBook Pro, if it's the older Thunderbolt Display, you'll need the same thing as your MBA - the Thunderbolt Legacy Adapter and the Thunderbolt Cable.  If it's the new UltraFine Thunderbolt Display, you'll need a quality Thunderbolt 3 Active Cable.
TL;DR
You can't do what you're asking.  The MBA only supports one display and while the MBP supports 2 external displays, it no longer supports Target Display Mode meaning, in this scenario, it only supports one display.  You might get the MBA to support two displays with another USB Video Adapter, but it will put a hurt on performance as USB is not an ideal interface for video.
My suggestion is to focus on your workflow to ensure all of your data is where you want it to be so that you can use any device, where ever you happen to be, and be assured that your data is intact.