No, it isn't safe, water bath canning is only safe for high-acid foods as the acid kills botulism. Low-acid food must be processed at 240F, 116C, and that can only be achieved in a pressure canner.
When you pressure cook the soup it kills the bacteria, however when you then transfer it to the sterilized jars it could be contaminated on the way, and then the water bath won't be hot enough to kill the bacteria.
Tomatoes aren't high acid, so they need the addition of vinegar or lemon juice in order to safely can with a hot water bath. Honestly, I'm not sure why you would want to make tomato sauce from canned tomatoes because for me, the whole point of canning tomatoes is because the tomatoes will otherwise go bad. But anyhow...
I recommend finding a tomato sauce recipe from a canning cookbook or recipe website. Here is a recipe I found from the Ball website which is a reputable source, and here is an article about hot water bath canning tomato sauce. Notice that the recipe calls for a high acid ingredient, which is extremely important in canning tomato sauce with a hot water bath!
Secondly, it is NOT okay to seal jars by inverting them!!!! Get yourself a book about canning, Ball sells several good ones (Blue Book and Complete Guide to Home Preserving) and learn about the canning process. The steps are as follows:
(0) Follow the recipe EXACTLY, don't add any other ingredients that may change the pH of the final product!
(1) Get the hot water boiling in a jar large enough to have a few inches both above and below the jar that you'll use. You can use a metal trivet or canning rings to act as a base for the jars so that they have water going under them and aren't touching the base of the pot directly.
(2) Start with jars that are appropriate to canning and have fresh lids (or clean, working reusable lids like Weck jars or Tattler lids). If processing for under 15 minutes the jars need to be sterile (can be made sterile in a hot water bath). Inspect Tattler seals for any nicks, and throw away the bad ones.
(3) Normal canning lids and Tattler lids need to be prepared by soaking in hot water right up until use.
(4) Add the ingredients to your jars, remove air bubbles with a spatula, wipe the rims clean, and apply the lid as per the instructions of your lid. (Regular Ball lids are slightly different from Tattler lids, and I've never used Weck jars which I think are different again.)
(5) Insert into the hot water bath with canning tongs, cover, make sure that you start the countdown once the water has returned to a full boil.
(6) Remove once the time is up. Don't tilt the jars when removing them as the seal isn't airtight yet. Check the instructions for your lids, at this point Tattler lids have to be tightened fully.
(7) Leave the jars alone to cool. After a few hours or overnight, check the seals by trying to gently pry open the can. They should be "finger tight" and not pop off. The jars that aren't properly sealed can be put into the fridge and consumed within a short time. Otherwise they're good for up to a year or whatever the recipe suggests.
Best Answer
The reason that people talk about "proper" trusted recipes in canning is that with anything less than that, all bets are off. Might keep, might not.
That's definitely true of your suggested process: they might keep in the fridge that long, might not, and as a bonus, if you get unlucky, there might be nasty undetectable botulism in there! Tomatoes aren't quite a low enough pH to avoid that, boiling isn't a high enough temperature to kill it either, your process won't even seal the jars, and refrigerator temperatures don't stop it from growing. You're really just using the jars as containers; there's nothing about the process that's actually "canning". You'd get the same results from putting tomato sauce in plastic containers.
And, honestly, a proper recipe is not even much more work than what you're talking about. For example, here's a trustworthy recipe for canned crushed tomatoes, pretty similar to what you want. It's pretty much:
So basically, that's your process with the addition of acid, and boiling the jars after filling, and it will make them keep at room temperature.
They have a bunch of variations if that's not exactly how you want them prepared.
If you aren't willing to put in that much effort, then I'd really suggest freezing. That removes all of the safety concerns with no complications at all. You can just prepare however you want, put into hopefully smallish containers so you can get at a manageable amount without thawing, and pop in the freezer.