There should be holes on the outside of the tracks to allow for drainage. Could be the hole(s) are blocked. If that is the case, it should be an easy fix.

If the track is filling up with water tells me the outside drain holes are plugged.

This is the most likely problem...

If the track drain holes were plugged, I would think that the contractor would have checked that if he was out several times?

Who is the contractor ?

I am having the same problem. They have been here 5 times to repair and on Saturday it leaked yet again.

Leaks
 

We had a similar problem, but once we had a porcelain tile floor installed, no more leaks. That blocked off any water from entering, and floor is completely in tact.

Agree with those who said the problem is drainage. Water is collecting in your track faster than it empties. This is either from blocked holes in the track, water flowing outside toward your home creating an external ponding that is higher than your drain holes, or possibly your roof alignment is dumping a huge amount of water down the outside of the window overwhelming the ability to empty.

Is it possible the cement ledge on which your sliders sit is slanted inward? Seems unlikely but worth checking.

Would be good to know for people considering florida rooms.

We worked with All Seasons Rooms. We were very satisfied with them during installation but have become disillusioned about their inability to stop the leak. They have not responded to our latest email (sent 6 days ago) and feel like they hope we go away. We may be left with no choice but to try and find another contractor who can fix it. We’re not big fans of legal remedies but have no problem with BBB and Seniors Against Crime.

Grade problem
 

Your house was built with the floors of the house above grade. Your lanai was built as an outdoor space and the floor is not above grade, but on grade. To expect the slider rails to keep out the flood of a heavy rain is asking a lot. You should have raised the floor. If not, I don't see you have a case.

Our lanai and bird cage floor was concrete tiles. Naturally, the rain runs through the spaces between tiles and comes into the lanai with heavy rains. The grade is away from the house so that keeps all but the heaviest rains out.

Do you have gutters in the back?

Maybe adding gutters to move the water away from that part of the house would help...

Lanai leak
 

We had the same issue, there is the main hardware attached to the house, then the secondary track (the windows sliders). The problem we had on both houses, was the caulking between the 2 pieces was not done properly. Only noticed when it rained hard. Both times they came out and recaulked between the 2 channels. This fixed the issues. The water will follow the track and leak at the point with no caulk.

French well
 

Try putting in a french well around the lanai or put in solid window about 15 inches high with shorter slides above.

If what you are saying is correct, and have no idea, would think the contractor should have brought that up to the buyer. Who wants a wet floor?