are TER tickets usable on a route anytime on the day they are valid?

That is why I never purchased a TER ticket on-line, always got it at a train station the day before or that morning, say Paris to Caen, or Paris to Beauvais

If you buy at the station they are good for any TER that day and route anytime.
You do not have to commit to a time to go or return.
I'm not sure about online.

When you buy a TER ticket online, you get instructions to retrieve your ticket from an SNCF ticket machine with the credit card used to purchase it, or at any ticket window when presenting said card. Then you validate it in the platform stamping machine before boarding the train. So when getting a round trip ticket, I believe you get two tickets, one for each way. The act of validating starts the clock for usage.

My memory is a little foggy on this, but did similar a few years back going from Beaune to Geneva with a connection in Lyon. Instead of taking the 1 hour 50 minute connection time, we took 3 hours and 50 minutes and toured Lyon for a bit before taking the next train. I can't remember if we got one or two separate tickets, but it was clear we could take any connecting TER train to complete the journey that day.

I've had several puzzling SNCF experiences while bouncing around France this summer. I check schedules and prices online, then generally use a ticket machine a day or so ahead of time for TER tickets. On one occasion I was ready to buy tickets for my next three trips (one of which included a TGV leg), and the vending fares were higher than I had seen online about an hour earlier. I opted to buy from one of the staffed counters, and the agent was very helpful in finding me decent prices and in one case a better schedule than I had been able to turn up. But as he handed me the tickets, he told me they were e-tickets that didn't need to be validated. They did, indeed, show specific dates, times and train numbers even for all-TER itineraries.

I've also had the SNCF website show different fares for different departures (all TERs) for multiple direct TERs on the same day (duration of trips virtually identical) when the vending machine gives only one fare for the usual seven-day period, and that fare is different from anything seen online about 30 minutes earlier. That was on a trip from Normandy into Brittany, so I now wonder whether some of the anomales I've seen were related to crossing regional boundaries.

I was also sold a ticket for same-day travel in a train that wasn't going to run that day, so I suspect there are some glitches in the processes used to update the schedule database.

I'm using a Carte Senior+, which may be the cause of these oddities.

The websites will list journeys for information purposes, and ask you to pick a specific one because of their programming, but TER tickets can be used for any train.

I found out or learned on this trip at Gare de l'Est that SNCF machines are useless to Americans since only the chip and pin credit card is accepted. That is stated in small print on the machine, and that one cannot pay in cash using that machine. That is shown in the pictogram, ie, you have to go to the staffed counter and wait in line.