A few things I would consider when looking at potential reinforcement.
This is all about deflection - not about a tank crashing through the floor. If you have enough deflection, your tank will typically fail long before you have a catastrophic failure of the floor joist, beams, etc. Split seams, cracked glass, damage to your stand, all possibilities if you have too much unwanted deflection.
You have several areas where there appears to have been water damage. Nothing to panic about, but it will make any structural calculators useless to even get a ballpark figure (which is all they can do on older construction). You can play with them, but it will be difficult since most as plugging in OSB, or 3/4" ply for the deflection calcs.
Also keep in mind many contractors will be concerned with overall structural failure of the floor - not taking into account a small deflection on a tank this size is really what you need to worry about.
So what can you do.... Here is what I would do before calling any contractor. Simple, sounds stupid, but put as much weight as you can in the spot you will be placing the tank. Then have a couple friend jump up and down right in front of the weight. "Feel" the deflection, is it spongy, do you see the pics on the wall vibrate, can you see the weight moving around?
I know this sounds incredibly un-scientific, but as a structural engineer, I can tell you not one person can reasonably calculate the exact deflection based on the water damage, age of wood, quality of qood, etc. We can easily come up with a 95-98% confidence level that you floor will not fail, but that is not what you care about.
While checking for deflection, if anything is moving, if anything is spongy, if the pics on the wall just vibrate any, you have a serious issue for a tank that size. Not saying an issue for a 55g tank - but an issue for a tank this size and as long as it is.
We just don't know enough about the structural setup to give you perfect advice so I will give general ideas to help guide you to a potential solution.
First, you need to be thinking about reinforcing on the actual floor, and underneath in the basement. They are both going to be critical if deflection is noticeable. If this is possible - I would laminate 2 maybe 3 pieces of 3/4" or greater plywood up on the first floor directly under the tank. This could be cutting the carpet back, removing the tile, etc. But you would want to get to the base floor. Using construction adhesive and rated screws to the joists, you will install each layer. This would be a minimum on the first floor - assuming you have deflection. Perfect world - you would do the entire room with at least 1 sheet, glued and screwed.
From the basement, there are too many unknowns to give a great answer. But I will tell you any work I do in a basement, I would want to see adjustable supports. Yes, a 2x6 wall is a great support structure, but seeing the water damage, I am concerned if that is recent enough to cause yearly movement of the wood - meaning swelling or contraction. The adjustable posts will take care of that.
If you go a beam route - use an exterior rates laminate beam, or a steel beam in a basement. Again, typical construction could care less if something contracts a 1/4", but your tank will require tighter specs. Basements are notorious for moisture fluctuations, so plan for that.
I know this sounds like another typical engineer over engineering something - and I totally get that. If this was a 90g tank and no signs of water damage, I would give you different advice. But there is a lot going on here and you need to take some time to gather all the advice and make a decision.
