What level find familiar is cast at depends on how the warlock casts it.
When she gains the Pact of the Chain class feature, the warlock learns find familiar. It doesn't count against her spells known, but nothing in the ability text prevents her from casting it with her spell slots. If she casts it with her spell slots, it works exactly like any other spell she knows how to cast. If she is 5th level when she casts it, she casts it as a third-level spell.
In addition, the warlock gains the ability to cast find familiar -- and only find familiar -- as a ritual. When a spell is cast as a ritual, it always uses the normal spell level of a spell. When the warlock casts find familiar as a ritual, she casts it as a first-level spell no matter what her warlock level may be.
The Warlock's spell slot level indicates the maximum spell level they may select.
I agree the wording is a little confusing but consider the general rule for spell slots:
When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell. You can think of a spell slot as a groove of a certain size--small for a 1st-level slot, larger for a spell of higher level. A 1st-level spell fits into a slot of any size [...]
In other words, the phrase "for which you have spell slots" more fully means "for which you have spell slots that the spell can fill."
The rules for Warlock spells known suggests newly gained spells for a Warlock may choose spells that are no higher than what is shown on the table for their level:
The Spells Known column of the Warlock table shows when you learn more warlock spells of your choice of 1st level and higher. A spell you choose must be of a level no higher than what's shown in the table's Slot Level column for your level.
Additionally, when you gain a level in this class, you can choose one of the warlock spells you know and replace it with another spell from the warlock spell list, which also must be of a level for which you have spell slots.
The wording here explicitly states that newly learned spells must be "no higher" than the slot level indicated for your current level on the warlock table.
The subsequent paragraph goes on to say that if you swap a spell out for a new one, that new spell must "also" be of a level for which you have spell slots. The word "also" indicates that it follows the same rules for the new spells (ie, equal to or less than the slot level on the table).
Another way of thinking about this: Classes that must prepare spells from a larger list (Druid, Cleric, Wizard) are told "The spells must be of a level for which you have spell slots." It's the same "for" wording, but no one would argue that these classes must prepare spells of equal (base) level available to them. They are free to choose only lower level spell and upcast them using the available higher level slots.
Altogether, the rules indicate that a Warlock is free to choose any spell that is of a level that is equal to or less than the slot level indicated on the Warlock table, regardless of whether that spell is swapped in or learned in addition to the previously known spells.
Best Answer
Some clarification first.
Warlocks
Does the Pact of the Tome Warlocks learn new ritual spells from other classes?
No
You can only get 3 cantrips as stated. Another reason why it does not allow you to research new spells is because of the Incantation "Book of Ancient Secrets", and that send us to the following point.
Book of Ancient Secrets
From the description (PHB 110):
and
With "Book of Ancient Secrets" invocation you can learn new ritual spells. These spells can only be cast as rituals.
Research
There are two meanings to the word "research" referent to wizards and spells; the homebrew type of research, where it is implied that you create new and custom spells, and the to add new (but existing in the game) spells to the wizard spell book.
Homebrew research
This type of "research" allows to create custom spells, as in literally new spells and modifications of existing spells. In the first instance you create spells based on an idea or an existing spell from other editions or games. In the second instance you take one spell, let say magic missile, and modified it to be more fun or align with a thematic; e.g. "Elemental magic missile". There exist guidelines in the DMG for homebrewing spells and has to be worked with your DM.
The way you create custom spells vary from table to table, some might just add to the wizard spell list, other would ask you to use your downtime time to "research" the spell and add to the list, or it can be the reason for an entire adventure. On the last point, just to add a bit of history, there are prominent wizards (that might or might not be relevant or exist in your game) through the history that left their work for future generation of wizards, two of the most common examples are Mordenkainen and his disciple Bigby.
Typical Research
From PHB 114, emphasis mine.
This segment feels like a typical day for a researcher. You pick an article (and if interest you), some times you need to decipher the notation, you keep reading and reading until you are sure that you understand the article, and then you transcribe it into your own article (or thesis). If in doubt of the results, you do the often undervalued action of trying to reproduce the experiment.
Downtime research
In PHB page 187 it is describe the action of researching as a downtime activity, emphasis mine. This can be ruled as a way to create homebrew spells as a mean of "research", or use the downtime to find spells and transcribe them in you spell book. This is up to the DM, though.
Note: Store playing
Bear in mind that store playing, as in Adventure League, might be against of homebrewing since they try to keep thing as balance and uniform as possible so you can move your character from table to table.