Train tickets-usable anytime same day

Yes only on the regional TER trains. These are the ones where you have to validate the ticket in the machine before boarding anyway or if it is an e-ticket the conductor will punch a hole it or scan it. It will be spelled out on this type ticket something to the effect: "Valid for travel anytime on this date."

Thanks for the information.

There is a complication. The TERs have two pricing periods, blue (off-peak) and white (peak). Although I had a senior travel card, it's my understanding that the distinction applies to non-discounted tickets as well. It's sort of like the rush-hour pricing we have on the Metro here in Washington, DC. I was told the timing refers to the departure time of the train. There's a copy of the calendar posted on each ticket machine, but it's really, really tiny and sometimes quite difficulty to read.

The fare calendar I found online is from 2016, which is unfortunate because there are some special situations during holiday periods, but at least it will give you an idea of what I'm talking about. Keep your eyes peeled as you walk though your first station or two. Usually you can find a card or brochure containing the current year's calendar in a rack. If not, ask at the information desk.

Oh, just found the current fare calendar, buried on page 122 of this document.

Of course, you can always just buy a ticket for the white period and not worry about it. On very short trips, there's not a great deal of difference. For longer journeys, it will waste some money if you end up traveling during the blue period.

If you have a TER ticket there seems to be different rules depending on whether you have an e-ticket or one you got at the station (either purchased there or picking up a print-at-station ticket)

TGV and Intercités tickets with a seat reservation + TER and Intercités E-Tickets
You can only take the train that you have reserved i.e. the one which you have been issued a ticket for. If you wish to depart earlier or later, you can exchange your ticket or cancel it and re-book it, depending on the fare conditions. Once you have done this, you can then choose a new departure time for your ticket.

Intercités and TER print-at-station tickets without seat reservations
You can take the same type of train, travelling the same route, at any time departing within 7 days of your reserved date. Please be mindful that certain fare conditions do restrict you from taking peak hour trains

https://faq.trainline.eu/article/146-change-train-departure

I can tell you from previous online purchases that a TER e-ticket does have a specific date and time on it so from the above it sounds to me like you need to take that exact train, even though you can sit anywhere you want.

If you are fluent in French you can try taking a different train and discussing it with the conductor ;-)

Thanks for the additional detailed information. I am familiar with peak and non-peak times from traveling on UK trains extensively.
I'll be sure to ask or read the fare calendar.