Since you are seeing analogue Ceefax, you have not yet been subjected to analogue swich-off. You are viewing analogue signals on both sets. There should be no modification of the displayed picture content as part of the analogue detection and video processing, therefore the picture should be the same on each set. I have just checked the Ceefax on my Samsung TV, never having previously looked at analogue on it. On the default page (100) there are four coloured headings at the bottom of the page which, from memory, are Sport, Weather, London TV and Index.
Is there a setting which defults the opening Ceefx page to other than 100? This is the only explanation I can think of for your experience.