non-linear DC-only bipolar model
These models come from two of the original three equivalent non-linear level-1 models of the bipolar transistor: the hybrid-\$\pi\$ (note: \$\pi\$-model) and the transport (note: \$T\$-model) versions.
The non-linear models are full-blown models that describe the behavior over all possible circuit arrangements. And because they are highly non-linear, they require non-linear solutions and are therefore rather complicated to apply.
Such complex, non-linear models can be linearized by taking their derivative and instead only looking at the behavior near some known operating point. This is the so-called quiescent operating point for a bipolar transistor circuit.
Taking the models from the above-linked answer:
simplified non-linear DC-only bipolar model, without leakage currents
When operating the bipolar transistor at a reasonable quiescent operating point, the reverse-biased BC junction only contributes a negligible leakage current, which can be ignored entirely as it is many, many orders of magnitude smaller than the forward-biased BE junction's current.
Before applying derivatives to create linearized models, the above non-linear models can first be reduced to:
And at this point, the familiar \$\pi\$ and \$T\$ shapes appear. Keep in mind these are equivalent models, mathematically. Just two different ways of expressing the same thing.
linearization concept: \$g_m\$
Now, the diodes in each of the above non-linear models will have a curve that looks something like this (taken from Desmos:
The red line shows the tangent slope at the operating point of \$V_{_{\text{BE}(q)}}=680\:\text{mV}\$ and \$I_{_{\text{C}(q)}}\approx 25.2538\:\text{mA}\$(for a model I chose where \$I_{_\text{S}}=10\:\text{fA}\$ and \$V_T=\frac{k\,T}{q}=25.9\:\text{mV}\$). Using this red line is what is meant when using the linearized model. That line is linear and we can use it to estimate the behavior of the bipolar transistor if and only if we don't stray too far from this assumed operating point.
Note that this red line has a slope, \$m=\frac{\Delta y}{\Delta x}\approx 97.5\:\text{mS}\$. This is also where the idea of \$g_m\$ arrives, as we'll see shortly.
But don't forget that this slope/idea only works, in this case, near \$V_{_{\text{BE}(q)}}=680\:\text{mV}\$ and \$I_{_{\text{C}(q)}}\approx 25.2538\:\text{mA}\$. Any substantial deviation from this operating point increasingly invalidates the linearized approximation.
linearization concept: \$T\$-model and \$r_e^{\:'}\$
From the transport model, without leakage included and keeping in mind that \$\beta_{\small{F}}=\frac{\alpha_{\small{F}}}{1-\alpha_{\small{F}}}\$ and \$V_T=\frac{k\,T}{q}\$:
$$\begin{align*}I_{_{\text{D}(q)}}= \frac{I_{_{\text{C}(q)}}}{\alpha_{\small{F}}}&=\frac{I_{_\text{S}}}{\alpha_{\small{F}}}\left[\exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)-1\right]\\\\\text{d} I_{_{\text{D}(q)}}&=\text{d} \left\{\frac{I_{_\text{S}}}{\alpha_{\small{F}}}\left[\exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)-1\right]\right\}\\\\&=\frac1{V_T}\cdot \frac{I_{_\text{S}}}{\alpha_{\small{F}}}\cdot \exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)\:\text{d}V_{_{\text{BE}(q)}}\\\\\text{But }\frac{I_{_{\text{C}(q)}}}{\alpha_{\small{F}}}\approx \frac{I_{_\text{S}}}{\alpha_{\small{F}}}\cdot \exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right),\text{ so:}\\\\\text{d} I_{_{\text{D}(q)}}&=\frac1{V_T}\cdot \frac{I_{_{\text{C}(q)}}}{\alpha_{\small{F}}}\:\text{d}V_{_{\text{BE}(q)}}\\\\\therefore r_e^{\:'}&=\frac{\text{d}V_{_{\text{BE}(q)}}}{\text{d} I_{_{\text{D}(q)}}}=\alpha_{\small{F}}\cdot\frac{V_T}{I_{_{\text{C}(q)}}}=\frac{\alpha_{\small{F}}}{g_m}\end{align*}$$
Note that I chose to use \$g_m=\frac{I_{_{\text{C}(q)}}}{V_T}\$, above.
So the diode in the transport model can be replaced by the above resistor because that resistor represents the voltage-versus-current slope of the diode at the quiescent operating point.
(This is the slope behavior of the diode at the quiescent operating point.)
linearization concept: \$\pi\$-model and \$r_\pi\$
From the hybrid-\$\pi\$ model, without leakage included:
$$\begin{align*}I_{_{\text{D}(q)}}= \frac{I_{_{\text{C}(q)}}}{\beta_{\small{F}}}&=\frac{I_{_\text{S}}}{\beta_{\small{F}}}\left[\exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)-1\right]\\\\\text{d} I_{_{\text{D}(q)}}&=\text{d} \left\{\frac{I_{_\text{S}}}{\beta_{\small{F}}}\left[\exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)-1\right]\right\}\\\\&=\frac1{V_T}\cdot \frac{I_{_\text{S}}}{\beta_{\small{F}}}\cdot \exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right)\:\text{d}V_{_{\text{BE}(q)}}\\\\\text{But }\frac{I_{_{\text{C}(q)}}}{\beta_{\small{F}}}\approx \frac{I_{_\text{S}}}{\beta_{\small{F}}}\cdot \exp\left(\frac{V_{_{\text{BE}(q)}}}{V_T}\right),\text{ so:}\\\\\text{d} I_{_{\text{D}(q)}}&=\frac1{V_T}\cdot \frac{I_{_{\text{C}(q)}}}{\beta_{\small{F}}}\:\text{d}V_{_{\text{BE}(q)}}\\\\\therefore r_\pi&=\frac{\text{d}V_{_{\text{BE}(q)}}}{\text{d} I_{_{\text{D}(q)}}}=\beta_{\small{F}}\:\frac{V_T}{I_{_{\text{C}(q)}}}=\frac{\beta_{\small{F}}}{g_m}=\left(\beta_{\small{F}}+1\right)\:r_e^{\:'}\end{align*}$$
Summary
The above uses two equivalent DC Ebers-Moll models to develop two different, but also equivalent small-signal linearized models based upon a quiescent operating point for a given circuit.
Note that the \$T\$-model's \$r_e^{\:'}\$ is the same as the \$\pi\$-model's \$\frac{r_\pi}{\beta_{\small{F}}+1}\$. There is some confusion over these two different model approaches. The \$T\$-model's \$r_e^{\:'}\ne\frac1{g_m}\$. Rather, the \$T\$-model's \$r_e^{\:'}=\frac{\alpha_{\small{F}}}{g_m}\$.
supporting documentation
I developed the above, directly from the level-1 DC Ebers-Moll models without referring to any documentation or attempting to refresh my memory. It's just math, after all.
But I just took an additional moment to reach into Adel S. Sedra & Kenneth Carless Smith's "Microelectronic Circuits", 7th edition:
The same results are found in a well-known textbook on the topic.