It looks like you got one of the newer 2-way's commonly used by most cable providers. Standard 1GHz, 3.5db loss out of each output. Check the cable going out of the splitter and into the TV. Occasionally if the connectors were not put on correctly you can get "suck out", which is where the semi conductor and cable get pulled out of the connector thus causing on a partial connection to whatever it is connecting to.

In all honesty, your modem should have a dedicated line from your groundblock (box outside). In most cases it won't matter too much but I highly recommend it. Most Groundblocks have multiple splitters or even a single 4-way splitter. 4way's have outputs of -7db from each leg. Add that on to the splitter you got going to the TV and Modem and you've got weak signal.

The pic you took of your TV is called tiling and it is generally caused by a weak signal/noise. Make sure to check those cable ends and have the cable company come out to replace the splitter if thats where the problem is.