Travelling from Gare de Lyon to Waldorf Astoria Versailles.

Any taxi will take you to Versailles. The trip can take much more than 30 min on a weekday, though. Right now, at 11 AM on a Tuesday, it would take 1 hour, and possibly up to 1h15-1h30 at the peak of the PM rush hour.
There is no need to reserve a taxi but sometimes there is a bit of a wait at the taxi stand in Gare de Lyon, so if you are in a hurry you can book one with G7 website or app (if you book, the meeting point is in front of the Mercure Hotel), specifying your train info.

By train, you would need to take RER A to La Defense and line L to Versailles Rive Droite, then walk 15 min or finish with a cab (but there might not be cabs waiting at Versailles Rive Droite, the taxi rank is small). If you can afford to stay at the Trianon Palace, you can afford a taxi.

I wouldn't at all bet that a taxi would be the fastest way, given the roads these days (and the locations of Gare de Lyon and Versailles relative to each other). But it's of course an option.

We are sr. citizens, travelling with luggage. That’s why I am asking about taxis. I just don’t want to become an easy target for some unscrupulous taxi driver. We don’t speak French. Could one please tell us an approx price from the Gare de Lyon to Versailles? How does the meter work? Is that depend on distance or total time of a trip? Thank you very much!

Can't help with the cost but just be certain that you get an official taxi. The super aggressive what I call scam taxis outside the train stations mist be avoided or you will surely pay a hefty price. We had one kept trying to pull my arm to direct us to his cab and this happened at two of the train stations in Paris. Just be cautious.

Fares are based on time and distance. Why don't you ask the hotel for a private pick-up. That way you'll have a set price with no guessing.

Taxis really aren't scammy at Gare de Lyon, and I have never encountered touts there (unlike Gare du Nord). There is an official taki rank in front of the historical part of the station, well signposted, 2 min walk from half of the platforms and 5-7 min from the other half. Go to the rank, wait for your turn. Taxis have a meter, I would count on 50-75€ for the trip to Versailles depending on traffic.

But indeed your hotel, given its category, can probably recommend or even arrange a private pickup. More than 100€ would be excessive, though.

The private pick up will be around 150€ and probably more if arranged by the hotel. Consider arranging for a pickup with Shuttle Inter.

Also a senior -- very senior -- and I'd be seriously cutting down on luggage and taking the train. One of the things we have done recently is get moderate size backpacks as our carry on rather than the messenger bags we used previously. We then use them when we do 1 to 3 nights side trips by train so we are not hauling a suitcase. But with smaller suitcases and less weight, and a backpack for airline carry on items, the train is quite manageable.

Update.
Thank you everyone for a lively discussion. After all we took an Uber. 49EUR, one hour ride. Everything went smoothly.

Thanks for coming back with a report ! Hope you have a nice visit.