'Rely' is defined as 'be dependent on' or 'be the main pillar of' or 'depend on with full trust or confidence'.
I somewhat agree with what you say (not 'solely' which I retract), or to say it another way, rely does not mean 'rely on to the exclusion of all other senses', but rather a reference to relying on 'sight' for a Perception check means a Perception check, with sight being the core, central and fundamental sense relied on.
Which is
not correct. You just went through the definitions. Ignoring the third, which is about how people see other people (trust and confidence are not relevant here), the first two clearly mean that there are other factors involved, and the first clearly shows that it includes situations in which the end result may involve many other factors.
10 boards are needed for a bridge to hold my weight. If any of the boards are removed, and I walk across it, it will collapse. Does the integrity of that bridge
rely upon each and every board for it to carry my weight? Obviously, yes. That is the common definition of rely, and the common understanding of what it means. Does your car rely upon the steering wheel to be safe to drive? If not, I can take it off for you and you'll be fine to drive it, right?
A perception check that relies upon sight is one in which sight would play a factor.
Let's look at it this way.
In the REAL WORLD, I am going to sneak up on you. It is foggy outside. I start out hiding behind a wall when you are facing away from me. When I walk out from around the wall and try to quietly approach you, there are a few things that might happen. I might reach you before you detect me, but that isn't likely as I am about as stealthy as an elephant. You might hear me before I reach you. You might turn and see me before I reach you. You might smell my manly musk before I reach you. There are lots of ways that individually I could first be detected.
But the
chance I am going to be detected is going to be a factor of all of those things, and we'll be relying upon all of those senses to determine the chance I will be detected. A perception check, which checks the chance of detection, relies upon sight unless sight plays no part.
I treat sight as playing no part when: There is total cover / heavy obscurement / blindness or an election is made to focus on another sense. However, if someone shuts off a sense, I might alter the DC (or grant advantage on an oppositional roll) if someone elects to not use their sight when they would otherwise benefit from using it in the detection.