No, 100% gauranteed your problem is the actual contacts in the starter. I have seen this same problem on my Supra, 4 Runner and many other Supras running around here in SA. It is VERY common on late model Toyotas. The brass contacts inside the starter wear down and do not make contact. Do not let anyone tell you that you must purchase a new starter. Simply go to Toyota and ask for a "starter rebuild kit". (not sure of the part #, but they have seen this problem many times) It is about $10 in parts to rebuild your starter.
It is a rather easy job to complete except there is this one little bolt that you will have a difficult time getting off. Not gonna tell you which one it is......you'll know it when you go to remove the starter!
Good luck and TRUST ME... it is the problem!
Thomas Kenyon