I didn't say anything about the wiki before, but I thought it would be easier to discuss if we sticked to the same proof/equation. If you looked at it like any other geometry problem then t would be the amount of time it takes for v or c to travel a certain distance. I think you should be able to solve for c in the proof by only using t and the length of the sides. It turns out that if you do that in the current settup you get the wrong value for c. I don't agree that a larger value for t should imply that time is going slower. I think a larger value for t would mean that there are more ticks on a clock, for more ticks to happen time would have to go faster. The distance the object traveled is ticks of the clock times velocity.
The equation finds the relation between the times of each side, it doesn't consider how many times the photon goes to the top and bottom of a clock. So then the answer should give you a direct translation of how much time has occured in one frame and give you how much time has occured in another frame, since c is the same on two sides.
I am sorry you have not concinced me yet, because I have gotten really good at algebra. They made me retake it twice from switching colleges and then I took it in high school and junior high 3 times. I got an A in every course, if they hadn't have done that we may not have had this problem... The theory here just doesn't seem to flow like older well done proofs. I apoligize for not being that forum savy.
Also if you do short substitution for the time dialation equation you don't get the length contraction equation. If you substitute t' in the equation L'=vt', then say L=ct you get the wrong relation between the two equations to gamma, and v≠v'. If a different value for v is found then put back into the equation it doesn't give the same value's for L and t. But, both observers should agree on the relative velocity.