Trainer tire selection (hybrid bike) I've been lurking for a couple months, but this is my first post and I still have a lot to learn.
Background
I inherited an old hybrid bike (Raleigh Dash EX) and an old stationary trainer, and I would love to find a suitable trainer tire to keep biking inside while it's cold and snowy.
This is my first time picking out a tire myself, so I need some feedback because I'm struggling to answer this question myself.
Here’s my rim (it's a 700c (622)), and the current tire (700 x 38): 
What's causing some concern for me when picking out a trainer tire is the tire width.
When I've looked at trainer tires, they tend to be much narrower. Examples:

Vittoria Zaffiro Pro Home Trainer Fold Tire: 700 x 23
Tacx Trainer Tire: 700 x 23
Kinetic by Kurt: 700 x 25

The only one I can find of reasonably close width is the Continental Hometrainer Folding Tire (700 x 32).
Question
Does this mean that my best choice of training tire is the Continental because the others are so far off in width from my current tire?
Another way to phrase this agnostic of endorsing a brand/specific tire: If a tire with width 38mm currently works well with my wheel, would it be better to find a trainer tire with a similar width (e.g. 32+ mm) or does the width not matter enough here to make an impact?
I've seen some different advice involving this because 1) if I were on the road, that width difference has the potential to cause rim damage (source), but 2) because I'm on a trainer, the tire isn't really taking much load so it being a perfect width might be less of a concern.
Follow-up related question: I'm assuming with such a big tire width difference from the existing tire, it's likely that I'll need to assess if I need a new tube. Is that a reasonable assumption?
Thank you!!
What's the width of the rim? (Measure it.) So long as the trainer tire is roughly the same width as the rim, or wider, you'll be just fine. You don't have to worry about hitting the rim or pinch flats as you do outside.