In the top block which lists your abilities, the black/white mark in front of the ability shows your bonus (and proficiency) for Saving throws. It says so under the block.
So when you become the target of a spell that requires a Strength Save to defend against, that's where you look up your bonus to the roll. In your case, it's +5. The black mark in front of Strength shows that you have Proficiency in Strength Saving throws, which means the +5 is made up of your proficiency bonus (probably +2) and your Strength (probably +3).
On the other hand, if you had to make a Wisdom Save, you would not add your proficiency, since there is no black mark in front of it. This means you just add your Wisdom modifier to the roll (the +1).
The lower block shows you what you roll when you make Skill checks. These are not defense rolls, but attempts to accomplish something in the game. Each skill is linked to an Ability. If you have proficiency with the skill, you add your Ability Modifier + Proficiency, if you don't have proficiency you just use your Ability Modifier.
See the difference between Performance and Persuasion; both are skills that are based on Charisma (which gives you +2 to the roll), but because you are proficient in making Persuasion checks, you get a +4 total bonus there since you're adding in your Proficiency.
Note that all the totaled numbers in both blocks have already been calculated. If the DM asks for a roll, just look up the number in the correct column/row and add (or substract) it from your D20. You don't need to add or remove anything from these numbers during play.
On your particular case, I don't think it is game-breaking to allow multi-classing at INT 12.
Since you are considering multiclassing for roleplay purposes, it makes sense to allow your character to get one Wizard level if you have INT 12. I don't see any particular issues with allowing a single level multiclass using the conditions you proposed. Your DM may rule that you cannot get further levels with only INT 12, since your Intelligence will stand below the average of a Wizard.
You might want to consider the Magic Initiate and Ritual Caster feats from the Player's Handbook (pages 168 & 169) as options to fulfill your roleplay as a "student of magic". Remember that you need at least INT or WIS 13 for the Ritual Caster feat.
Best Answer
Your question looks at the D&D 5E ability check system a little bit backwards. The mechanic is in fact called ability checks, not skill checks or tool checks, and it's intended to be used looking at the ability first, and then figuring out whether proficiency in a skill or tool might apply. From the Ability Checks section of chapter 7 (my emphasis added):
And then looking at how Tools work from Chapter 5:
So rather than looking at it from the perspective of "How do I do a check with cooking utensils?", we need to look at it from the perspective of "Why is the DM calling for an ability check?" There are plenty of things involving cooking utensils that probably wouldn't call for a check at all, since making a normal everyday meal while in a home with all the ingredients handy isn't really something that tends to have much of a chance of failure or much impact of success. But sometimes the character is trying to do something where it makes sense to use a roll to determine if it succeeds, so the DM would call for an ability check. And then if having proficiency with cooking utensils would be helpful to accomplishing the objective, then the proficiency bonus would be added. Here are some ability checks I can imagine calling for where one could reasonably add the proficiency bonus for cooking utensils:
As you've noticed, other than the general suggestions provided in the "Using Each Ability" section, it's really completely up to the DM when ability checks are called for, and which ability applies to the task that the character is trying to do. Different DMs definitely have different philosophies and approaches to it. And sometimes really more than one skill or tool proficiency could reasonably apply. I've certainly (as DM) called for Charisma checks allowing for either their Persuasion or Cooking Utensils proficiency to be applied, whichever they had, since they were trying to make friends using food. But once the DM picks which ability to use for the ability check for whatever it is that the character is trying to do that requires a roll, it's not usually that hard to figure out if proficiency in a tool might apply to the roll.
Note that since "Your proficiency bonus can’t be added to a single die roll or other number more than once," (Ch. 7, "Proficiency Bonus") even if you have proficiency in both a skill and a tool that might apply to the ability check, you still only add your proficiency bonus once. (And Blckknght in the comments says that there is an optional rule suggested in Xanathar's Guide that could grant advantage in that situation, but I don't have that book so you'll have to look that up for yourself.)