Toyota Repair: toyota repair question, cold crank, cylanoid

Question
I was happy to find this question forum! I am at witts end with my 92 Toyota corolla. It has 148,000 miles 1.6 liter manual tranny. 2 months ago, it would not start periodicaly. I took it in and they said I needed a new starter,, so they put one in. The car started for a day then wouldn't. Then the next day it did. for about a month it started then out of the blue, it wouldn't start again. It was a monday. I had it towed home, my dad tried to start it on tuesday, it started. so Wednesday I took it to work, wehn leaving work it wouldn't start. I thought bcuz of cold weather, so bought a bigger cold crank battery, it worked on thursday, but then would not start on friday. But when I had it jumped it would start. So i took alt to get tested and it was good. so we just bought new spark plugs, but they were good too. I am wondering could it be the cylanoid? does my car have one? PLEASE can you help? any ideas why the car is being so picky and not wanting to start? thank you!

Answer
The starter solenoid is part of the starter,was the replacement starter from toyota or an independent source, if so, I highly recommend replacing the starter with a remanufactured one from the toyota dealer, it sounds like the starter is causing the problem.