Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Anatomy A muscle that stretches or tightens a body part.
 n. Mathematics A set of quantities that obey certain transformation laws relating the bases in one generalized coordinate system to those of another and involving partial derivative sums. Vectors are simple tensors.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A muscle that stretches a part, or renders it tense.
 adj. Of or relating to tensors
 v. To compute the tensor product of two tensors.
from the GNU version of the Collaborative International Dictionary of English
 n. A muscle that stretches a part, or renders it tense.
 n. The ratio of one vector to another in length, no regard being had to the direction of the two vectors;  so called because considered as a stretching factor in changing one vector into another. See Versor.
from The Century Dictionary and Cyclopedia
 n. In anatomy, one of several muscles which tighten a part, or make it tense, or put it upon the stretch: differing from an extensor in not changing the relative position or direction of the axis of the part: opposed to laxator.
 n. In mathematics, the modulus of a quaternion; the ratio in which it stretches the length of a vector.
 In anatomy, noting certain muscles whose function is to render fasciæ or other structures tense.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. any of several muscles that cause an attached structure to become tense or firm
 n. a generalization of the concept of a vector
Etymologies
Examples

For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space).
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by ds2 = ρ2dθ2 – 2a sin2θdrdφ + 2drdu + …
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

EDIT: A note from The Unapologetic Mathematician that I should add: the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

A metric tensor is a function defined on a manifold (a vector space) that takes in two tangent vectors and produces a scalar quantity.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

The matrix representation of a metric tensor is a matrix.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

As in general relativity the metric tensor is written as gij = ηij + hij …
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

A metric tensor is NOT the same object as a metric, it is NOT the same object as its matrix representation, and it is NOT the same object as its associated line element.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

An old problem in tensor analysis is canonicalization of tensor expressions; telling whether one can swap indices around to establish equivalence or not.

A metric tensor is not the same as a metric (it’s more analogous to an ‘infinitesimal’ metric function), but it is usually understood in differential geometry and related areas in physics that when one says “metric”, they really mean “metric tensor”.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

As for the line element, there’s one thing you missed in your complaints: the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element
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