You need to make sure that your Lr catalog, i.e. the Lr database, i.e. the lrcat-file is on the external drive.
With Lr you always need two things: your photos (image files), be they Raw-files or not and your catalog (= database). The catalog contains all what you do in Lr; it also contains a reference to the file names and location of the photos (image files).
That means both computers need to have access to the image files and to the Lr catalog.
In Lr go >Edit >Catalog Settings >General tab. It tells you there the name of the catalog currently used and its location. Then in Win Explorer / Mac Finder move this catalog file to your external drive. Create a top-level folder on your external named "Lightroom catalog and Previews", so that you know where it is located. Then double-click the catalog file to open Lr with it, or - alternatively - open Lr (it will come up blank since you moved its catalog) and go >File >Open Catalog ... , navigate to the catalog file on te external drive and select it. From now on Lr will use this catalog. Do likewise on your laptop, i.e. direct Lr to the catalog on the external drive.
You would be wise to backup your photos and the Lr catalog to a different drive (kept safely at home) for safeguarding in case your external drive goes AWOL during travel.