1. "A tangent is a straight line which only meets the curve at that
one point."
Counterexample: curves with two or more stationary points can be used as a counterexample. An example of one of these is the cubic y = x^3-3x+3. The line y=5 is a tangent at the local maximum at (-1,5), yet it also crosses the line at (2,5). This means that it meets the curve at two points, yet is a tangent.
2. "A tangent is a straight line which touches the curve at that
point only."
The above curve can be used again for this definition, since it touches at two points. In addition, curves such as y = x^4-x^2 have two stationary points with the same y-coordinate, so these can have straight lines which are tangents to them at two points.
3. "A tangent is a straight line which meets the curve at that point,
but the curve is all on one side of the line."
The first example can be used here too.
4. "A tangent is a straight line which meets the curve at that point,
but near that point, the curve is all on one side of the line."
This depends on your definition of 'near'. For example, the curve y = x^3-x has a local maximum (with which a horizontal tangent can be drawn to) and another point with the same y-coordinate (which the tangent will go through, making the line on the other side of the curve) with a difference in x-coordinate by less than 2. However, for the curve y = x^3-0.001, which is similar but with a different coefficient for the x term, these two points are within 0.06 of each other, which is definitely 'near'.
A good definition would be something similar to: "A tangent is a straight line which passes through the same point as the curve, and at this point has a gradient perpendicular to that of the curve."