Thinking: The ability to arrive at results from the consideration of and choice from alternatives.
As stated, computers and very simple animals, and perhaps even plants, may think, namely if what they do involves some consideration of and choice from alternatives. Also as stated, this may be clearly quite mechanical, or explained in mechanical ways, without necessarily or always being mechanical, or reducible to some kind of primitive recursive function.
Furthermore, there is clearly the possibility to say, also with considerable etymological justification, that real judgment and real art are not reducible to rule or rote: Creative thinking involves more than automatical application of received rules to inert material, just as sound judgment involves careful discrimination, distinction, and appraisals of evidence, feelings and interests that again cannot be reduced to a mere logical application of case law.
And given these clarifications, there seems to be a wide spectrum of things that may be said to think in some sense, from calculating machines and computers, to plants, animals and human beings, and involving all manner of complexity, from the crudest automatic choice by the simplest rule based on a given condition, to a highly complex process of combined feeling and thinking and intuition, that nevertheless may be be necessary for and issue in the proof of a mathematical theorem or the composition of a symphony.