Not in the general case

While a parallel could be made between 'teleportation' and 'magic' such that only those things specifically called out as teleportation count as teleportation just like only those things called out as magic are magic, no such parallelism exists in the rules at present. Instead, 'teleportation', having no special meaning, is supposed to default to the "natural language" sense of the word, even though that is not generally a linguistically coherent course of action.

Since we are also supposed to do this because we are pretending that the rules themselves are written such that each word uses its single idiomatic meaning, it would follow that all things explicitly labeled teleportation are certainly teleportation, but other stuff might be teleportation as well.

Consequently, although we cannot say that you definitely make the Charisma save with spells like tree stride, we also cannot say that you can definitely bypass it. Instead, it depends on whether your DM decides the effect counts as teleportation or not, based on what they think the single universal meaning of the word in idiomatic English is.

Yes. Tree stride and transport via plants do not incur the charisma save to escape from a force cage

Both tree stride and transport via plants provide a location shifting effect, but do not state they are teleporting the caster. Spells that teleport the caster state they do so in the effect text.

The drawback is the plant spells are dependent on the presence of a tree or large inanimate plant inside the force cage's area. Clever use of a tree or mass of plants if they are available.

Teleporting spells

In contrast, spells that teleport the caster state this, e.g. " you teleport..." "You and up to five willing creatures within 5 feet of you instantly teleport..." The following spells all indicate they are teleporting the caster and are subject to the charisma saving throw effect:

• misty step
• dimension door
• word of recall
• thunder step
• far step
• arcane gate

Availability

An import difference here is that the spells that afford magical movement that isn't teleportation are dependent on the presence of a suitable trees or plants whereas the direct teleportation spells are always available to the caster.