First of all, I would like to say that it is possible that these terms are used inconsistently, given that at least transfer learning, AFAIK, is a relatively new expression, so, the general trick is to take terminology, notation and definitions with a grain of salt. However, in this case, although it may sound confusing to you, all of the current descriptions on this page (in your question and the other answers) don't seem inconsistent with my knowledge. In fact, I think I had already roughly read some of the cited research papers (e.g. the MAML paper).
Roughly speaking, although you can have formal definitions (e.g. the one in the MAML paper and also described in this answer), which may not be completely consistent across sources, meta-learning is about learning to learn or learning something that you usually don't directly learn (e.g. the hyperparameters), where learning is roughly a synonym for optimization. In fact, the meaning of the word "meta" in meta-learning is
denoting something of a higher or second-order kind
For example, in the context of training a neural network, you want to find a neural network that approximates a certain function (which is represented by the dataset). To do that, usually, you manually specify the optimizer, its parameters (e.g. the learning rate), the number of layers, etc. So, in this usual case, you will train a network (learn), but you will not know that the hyperparameters that you set are the most appropriate ones. So, in this case, training the neural network is the task of "learning". If you also want to learn the hyperparameters, then you will, in this sense, learn how to learn.
The concept of meta-learning is also common in reinforcement learning. For example, in the paper Metacontrol for Adaptive Imagination-Based Optimization, they even formalize the concept of a meta-Markov decision process. If you read the paper, which I did a long time ago, you will understand that they are talking about a higher-order MDP.
To conclude, in the context of machine learning, meta-learning usually refers to learning something that you usually don't learn in the standard problem or, as the definition of meta above suggests, to perform "higher-order" learning.
Transfer learning is often used as a synonym for fine-tuning, although that's not always the case. For example, in this TensorFlow tutorial, transfer learning is used to refer to the scenario where you freeze (i.e. make the parameters non-trainable) the convolution layers of a model $M$ pre-trained on a dataset $A$, replace the pre-trained dense layers of model $M$ on dataset $A$ with new dense layers for the new tasks/dataset $B$, then retrain the new model, by adjusting the parameters of this new dense layer, on the new dataset $B$. There are also papers that differentiate the two (although I don't remember which ones now). If you use transfer learning as a synonym for fine-tuning, then, roughly speaking, transfer learning is to use a pre-trained model and then slightly retrain it (e.g. with a smaller learning rate) on a new but related task (to the task the pre-trained model was originally trained for), but you don't necessarily freeze any layers. So, in this case, fine-tuning (or transfer learning) means to tune the pre-trained model to the new dataset (or task).
How is transfer learning (as fine-tuning) and meta-learning different?
Meta-learning is, in a way, about fine-tuning, but not exactly in the sense of transfer learning, but in the sense of hyperparameter optimization. Remember that I said that meta-learning can be about learning the parameters that you usually don't learn, i.e. the hyper-parameters? When you perform hyper-parameters optimization, people sometimes refer to it as fine-tuning. So, meta-learning is a way of performing hyperparameter optimization and thus fine-tuning, but not in the sense of transfer learning, which can be roughly thought of as retraining a pre-trained model but on a different task with a different dataset (with e.g. a smaller learning rate).
To conclude, take terminology, notation, and definitions with a grain of salt, even the ones in this answer.