First off, the Achilles tendon is not a muscle - it is the common tendon that connects the "calf" muscles (gastrocnemius and soleus) to the calcaneus (heel bone). The calf muscles are indeed the plantarflexors (plantarflexion is the movement of pointing your toes away from you) - they are important in standing on your tip toes, in toe off during the gait cycle, etc.
However, the general consensus among the medical community is that it's now called
tendinopathy, not tendinitis. This is because there is no evidence to show activity of inflammatory cells, nor does it heal with the classic triphasic inflammatory response (aka the soft tissue healing model). Now, the importance of this is in the way it is managed - the word "tendinitis" makes us think of inflammation and makes us want to completely rest, etc. which is actually bad for a tendon. Won't go into the theory behind the pathology here, but always happy to share if you're curious.
You're right with the causes - a sudden increase in load is a common cause of tendinopathy. Tendons hate change, and they hate compression. So you'll also get tendinopathy when the tendon is excessively compressed (at the insertion, it is compressed in dorsiflexion (aka pointing your toes towards you) - ergo, this is where you get insertional Achilles tendinopathy). Tensile loading is also hard for tendons - and this is one of the loads that's more troublesome in mid-portion Achilles tendinopathy.
Tendinopathy is characterised by very localised pain - you're generally able to point with one finger to the specifically sore area. The pain is aggravated by dose dependent tendon load, and in the Achilles, you generally get aggravation from double or single leg calf raise, jumping, or hopping. You also generally get pain and/or stiffness in the morning.
In terms of treating tendinopathy - it's important to receive this from a physiotherapist, ideally one who treats a lot of tendinopathy.
According to Prof Jill Cook, and Dr Ebonie Rio (had a series of lectures from her these past two weeks!), and a host of other world class tendon researchers, we should not be using stretching in our treatment (puts tensile load on the tendon, which is bad!). In a nutshell, because obviously, not here to give medical advice, the stages of rehab are focused on load management (taking out the abusive load but still loading the tendon), then strengthening (lots of heavy isometrics!! Slow and heavy is the mantra), working on the tendon's energy storage, then working on that elastic function - energy storage and release (which is super important in normal function!).
Prevention - it's important to keep the exercises going, to improve the strength and function of your kinetic chain, avoid excessive compressive/tensile loads, and avoid dramatically increasing your load at the snap of a finger (do it gradually)
Happy to provide references here.