HDMI connection/ext1

First, we can see that your Sony does not have a TNT tuner: https://www.homecine-compare.com/lecteur-SONRDRHX680B-SONY-RDR-HX680B-160-Go.htm, which poses a recording problem. Indeed, you would need to buy a TNT tuner with a Scart output to connect it to the recorder. The TV cable will be connected to its antenna input and its TV antenna output will go to the Toshiba's antenna input.
As for the player itself, connect it via HDMI to the TV, switch the TV to that input (note the assigned number). For the Sony, you may have an "HDMI" button or a button to switch the audio/video output. If you manage to switch the output, go to the Sony's settings menu and adjust the resolution to 1080p and the audio to analog or PCM or stereo.
The simplest solution in the future is to keep it as a simple DVD player and get an HD TNT decoder that can record to a hard drive.
A simple example: an Akira DHB-B36HR https://www.homecine-compare.com/lecteur-AKIDHBB36HR-AKIRA-DHB-B36HR.htm, affordable here: https://www.topachat.com/accueil/index.php, a case for a 2.5" hard drive: https://www.topachat.com/pages/detail2_cat_est_micro_puis_rubrique_est_w_boidd_puis_ref_est_in10048028.html, and a hard drive, a 500 Go for example, but you can choose whatever you want: https://www.topachat.com/pages/detail2_cat_est_micro_puis_rubrique_est_wdi_port_puis_ref_est_in10051602.html. Initialize it with your PC and format it to NTFS. Connect it to the Akira decoder and reformat it to NTFS as well. The recorded audio/video files are in "ts" format and you will be able to watch them on a PC, a multimedia HDD, or directly by connecting to the TV's USB port (nothing is specified about its compatibilities).

Is your player connected via HDMI or SCART?
Old "tube" TV or new LCD?

The old TV was a Philips LCD flat screen and as I mentioned in my previous message, the connections are the same: an HDMI cable from the player to the TV and a SCART cable from the player to the TV (ext1).

Why two cables? One HDMI and one SCART? One or two players?

You need to switch the HDMI input of the TV to the connection of the player/recorder, before that check that the audio output is set to PCM as your TV may not be compatible with Dolby Digital and check the output resolution.
It would be good to communicate the brands and models of each device.

I forgot to mention that I am subscribed to Canal and that I receive the programs via satellite, which poses no problem for me to record, but by going through the SCART (ext1). If I forget to switch to the ext1 input when I program..... no recording.